Are quantum gravity theories trying to progress on the assumption that there are such things as gravitons?
We had this discussion a couple of times; it depends what you mean by gravitons - and it depends which formalism of QG you are using.
One problem with gravitons is that the whole concept of particles in quantum field theory relies on a a background metric which allows you to define fourier modes, Hilbert spaces with operators, propagators and particles. As it is expected that QG does not have such a background structure, the whole concept of particles and gravitons is expected to change.
In SUGRA you have a graviton field which is quantized and is rather close to a formalism with gauge bosons like the photon. In string theory the graviton emerges as one degree of freedom of the quantized string. In LQG the fundamental degrees of freedom are something totally different (spin networks) but it is expected that a graviton emerges as a low-energy limit of the theory, so it is not a fundamental degree of freedom.
