MTW has an approximation of the formula given above in a post by ergosphere, which is known as the quadrupole formula. See page 978, section $36.2, "Power radiated in terms of internal p ower flow".
In non-geometric units, MTW's formula, which is derived in the context of one body orbiting another is:
$$P_{GW} \approx \frac{ \left( P_{internal} \right) ^2}{P_0} $$
##P_{GW}## is the power (energy/unit time) radiated away by gravitational waves. It's a weak field approximation, so among other assumptions we are assumed there is no significant gravitational time dilation. The formula will work approximately for things like the Hulse-Taylor binary, it won't apply (nor will the quadrupole formula apply) in the strong-field regime of a pair of inspiraling black holes.
##P_0## is a constant, equal to c^5/G, G being the gravitational constant. Numerically, in SI units it's ##\approx 3.62 \, 10^{52}## watts.
The large size of ##P_0## means that the ratio of the ##P_{internal}## to this constant in the weak field regime is small, much less than one.
##P_{internal}## is the internal power flow of the system. It's described as the product of:
The mass of the part of the system that moves (in a circular orbit)
( Size of the system^2 )
( The reciprocal of the period of the system )^3
The argument uses Kepler's law (this is a weak field approximation!), which in this context is:
The ratio of the square of an object's orbital period with the cube of the semi-major axis of its orbit is the same for all objects orbiting the same primary.
That's why it involves the ratio of the square of the radius of the orbit and the cube of the orbital period. ##P_{internal}## is equivalent to the non-spherical part of the energy of the system, ##\approx m v^2## per unit time, i.e. the non-spherical part of the power of the system.
The units work out, the square of a power divided by a constant power is a power.
This is an approximation of an approximation. There are various constant factors omitted.
If you still want to understand the quadrupole formulation better, the ##I_{jk}## is rather similar to the moment of inertia tensor, if you are familiar with the moment of inertia tensor. Then some diagonal elements are subtracted to make the trace of ##I_{jk}## vanish, making it the so-called "reduced" quadrupole tensor. Finally, to use the quardrupole formula, we need to take the third time derivative of this tensor. And to get the total emitted power, we'd have to project the resulting tensor expression to various angles from the source, then integrate. The complexities make it a bit hard to understand the physics, the formulation in terms of internal power flow makes the physics a bit more understandable. However, if you are interested in cases other than GW's emitted by a circular orbit, it's unclear to me how well MTW's formula would apply.