We don't actually know that there is a consistent approach to quantizing General Relativity. What we do know is the following:
(1) in 2+1 dimensions, one can consistently formulate a quantum theory of "gravity". The reason for the scare-quotes is that no quantum theory in less than 4 dimensions can lead to the quantization of independent gravitational degrees of freedom -- because there are none. More precisely: the Weyl tensor (which contains the gravitational degrees of freedom) is 0 in less than 4 dimensions. Or, to put it another way: all quantum theories of gravity in 3 or fewer dimensions have c-number Weyl tensors (since the 0 tensor is a c-number).
(2) In 3+1 dimensions, the only known approach that has led to a quantization of Einstein's law of gravity was that devised by Carmelli in the 1980's. The most important feature of the formalism is that it is not cast in Riemannian geometry, but Riemann-Cartan geometry. The distinction is crucial because in it, the metric remains classical, while the connection is quantized as the connection of a gauge field (namely, an SL(2,C) gauge field).
The reason this has not been heralded as the Final Definitive Solution to the Problem is that it only works for purely gravitational fields. That is, if all you're interested in is the exterior solutions in a matter-free vacuum, Carmelli does the job. Unfortunately, Carmelli never found a way to even couple the classical theory with matter, much less the quantum theory.
The most notable feature of the theory is that the Weyl tensor is a c-number.
(3) The approach "Quantum Field Theory in Curved Spacetime" succeeds in formulating quantum theory in a general relativistic context. However, there are two main features that are both regarded as drawbacks (whether regarded rightly so as drawbacks, on the other hand, is itself a question for contention). First, there is no reaction of matter on geometry. Rather, the curved background serves to condition the propagation (the Greens functions) and the wave equation. Second, one needs to make restrictive assumptions that, themselves, cannot be framed in operator form in any theory that has the metric as a quantized dynamic variable -- namely, that the underlying spacetime be globally hyperbolic.
The global hyperbolicity assumption is not expressible in operator form. So in a prospective quantized theory of gravity, one could literally have a superposition of a globally hyperbolic state with one that is not. Unfortunately, since nobody's ever found a consistent way to do quantum theory in a globally non-hyperbolic setting (this is much of what the 1990's papers about time travel and closed time loops was about) then the situation could be likened with the worst form of a Schroedinger Cat: a superposition of a (globally hyperbolic) universe in which quantum theory can be defined, with a (globally non-hyperbolic) universe where it can't be.
The more basic problem is that even metric signature is not something that can be expressed in operator form. So, one could even have a superposition of a state that is a 3+1 spacetime with a state that corresponds to a 4 dimensional timeless space. Given how central the notion of time is to quantum theory, this seems to entail some serious consistency problems.
(4) Loop Quantum Gravity takes place in the setting of Riemann-Cartan geometry. It tries to adopt a "background free" approach, placing the diffeomorphism group at the center. Unfortunately, not all things relevant to physics are acted on by the diffeomorphism group, so that the very notion of background freeness itself can't be consistently defined. More precisely: the kinds of objects acted on by the diffeomorphism group are what mathematicians call "natural objects", and the corresponding operations are called "natural", while the underlying geometries are referred to as "natural bundles". The kind of geometry required for a theory of gravity that satisfies the equivalence principle is a subset of the tangent bundle known as an "orthonormal frame subbundle". I don't believe the orthonormal frame bundles are natural. This is closely tied to the problems raised at the end of (3). In addition, fermions require spinors. These reside on spinor bundles and I don't believe these are natural bundles either.
One way to approach this may be to relax (or redefine) the requirement so as to only require "gauge naturalness". In place of the diffeomorphism group Diff(M) on a spacetime manifold M, this broadens to the gauge group Gau(P) and automorphism group Aut(P) on a principal bundle P that has M as a base space. Gauge natural objects over P need not be reduce to natural objects on M. In addition, I don't believe gauge natural objects on P are natural on P (i.e. that they are not acted on by the diffeomorphism group Diff(P)). So, one has background structure by virtue of the confinement of focus to Aut(P) and Gau(P).
(5) Sardanashvily, Mangiarotti, et. al work off of a long-lasting strand that originated with Heisenberg and Ivanenko. They (rightly) point out that the reduction of the manifold M's tangent bundle TM to the orthonormal frame bundle F_g(M) associated with a metric g is a form of symmetry breaking. All this, of necessity, takes place in the broader setting of Riemann-Cartan geometries. This puts the spotlight on either the metric g or the frame fields as being the corresponding Goldstone-Higgs fields. In general, when you have symmetry breaking, the Goldstone-Higgs fields are essentially classical. Each different configuration corresponds to a different vacuum state and different coherent subspace; and between any two coherent subspaces no quantum superpositions can occur.
They highlight the issue of the fermions, noting that the very process of quantizing it, itself, critically depending on which subbundle F_g(M) of TM you choose; so that two quantizations corresponding to inequivalent g's must lead to different quantum state spaces; i.e. no quantum superpositions of the usual kind between different states can exist if the states disagree on whose motions are free fall/inertial.
Among other things, when symmetry breaking is present, the vacuum is no longer a unique state. So the premise of a unique state |0><0| underlying an equation such as <0|T|0> = kG (which the approach in (3) may use) is false, because the equation can't even be written down.
(6) String theory. I don't know enough to say anything about it.
(7) Arguments against classical/quantum hybrids, such as the famous Feynman argument tend to be premised on Riemannian geometry. When the arguments pass into folklore this tends to muddy the issue and lead to fallacious claims. In fact, the Feynman argument employs fermions, which require a Riemann-Cartan geometry. In a Riemann-Cartan geometry one necessarily make a distinction between a metrical extreme curve and a geodesic curve. The former is a "geodesic" for the Levi-Civita connection, while the latter (the *actual* geodesics) are geodesic for the connection native to the Riemann-Cartan geometry.
The Riemann-Cartan geodesics for two electrons in different spin states CAN diverge from one another, in the way that the Feynman argument visualizes.
Another issue, whose premise on Riemannian geometry is often forgotten, is the question of what equation would govern the determination of geometry from quantized matter. Here, the problem is that (in a Riemannian geometry) if the metric is classical, then so is the Levi-Civita connection. Therefore, the field equation would have a (classical) Einstein tensor G on one side and a (quantized) stress tensor T on the other. So, one tends to use a proxy, like <0| T |0> = kG, where |0><0| is the (assumed unique) vacuum state.
Given the issues previously raised about vacuum degeneracy, the premise of the equation is questionable.
If, on the other hand, the vacuum is degenerate with (say) one state |g><g| for each orthonormal frame bundle F_g(M) then it is quite feasible to write down an equation such as <g|T|g> = k G(g), where the right hand side is the Einstein tensor G for a given metric configuration g. The same question regarding how the matter is to be coupled to gravity still arises, but is partially negated in a Riemann-Cartan geometry since in such a geometry the connection is an independent object. So one could quantize the connection, while keeping the metric classical.
(8) Closely linked to this is the idea of gravity NOT being a fundamental force at all, but effective. This is advanced by Padmanabhan, Verlinde, et al.; and should also be linked to Sardanashvily, Mangiarotti et al. as well as to Jacobson's Gravity-as-Thermodynamics idea, which Verlinde descends from.
This is the approach I think is the right one. It can be deepened if one is able to derive an equation such as the one I posed <g|T|g>/h-bar = 8 pi A G(g) (A = Planck area) as something arising from an anomaly associated with a breakdown of classical symmetry. In particular, this could be something that comes about as a result of diffeomorphism symmetry being spoiled upon quantization.
(8) Classical/quantum hybrids.
All successful approaches I've described entail the same thing: the Weyl tensor is a c-number and the metric is classical -- even if the connection is quantized. This is not consistent in a Riemannian geometry, but is perfectly well feasible in a Riemann-Cartan geomtetry. The main observation is that the equations governing matter (particularly fermions) only see gravity through the connection, and only see it as just another gauge field. So, as long as the connection can be quantized, this part of the consistency problem is resolved.
The main issue with approaches to quantum field theory that adopt microcausality as a postulate is this. Since the axiom is posed at the operator level, as a defining condition on the field algebra itself, this has the effect of building in the light cone structure of WHATEVER geometry emerges from the algebra. But that sets into motion a chain of consequences:
(a) once you have the light cones, you have the conformal geometry
(b) once you have the conformal geometry, you have the Weyl tensor -- a unique tensor for each field algebra; i.e. a c-number tensor.
(c) once you have a c-number Weyl tensor, you have no quantized gravitational modes, since it's the Weyl tensor that defines the gravitational degrees of freedom.
So, eventually the correct approach to the problem will lead to an IMPOSSIBILITY THEOREM for Quantum Gravity (when "Quantum Gravity" is meant in the sense of "quantized metric" or "quantized Weyl tensor"), and the establishment of a quantum theory in which gravity emerges as an effective force, rather than as a fundamental force.
In such an approach, the item (3) "quantum theory in a curved background" IS all you have and all you need. The back-reaction of matter on geometry is embodied in an effective dynamics, such as the one I posed: <g|T|g> = kG(g).
Last, but not least, bearing on this question are the works of Penrose, Diosi et al., who have been seeking ways to hybridize classico-quantum forms of gravity.