In general it is either false or undefined that gravitational waves propagate at $c$.
As a counterexample, GR predicts that a gravitational-wave pulse propagating on a background of curved spacetime develops a trailing edge that propagates at less than $c$. See Misner, Thorne, and Wheeler, p. 957. This effect is weak when the amplitude is small or the wavelength is short compared to the scale of the background curvature.
For a general spacetime that includes high-amplitude gravitational waves, there need not be any meaningful definition of the speed of propagation of such waves. It's only in the low-amplitude limit that we can meaningfully talk about splitting a metric into a background term and a wave term. In general such a split is not uniquely defined, because there is in general no notion of adding metrics the way we add fields in SR. To add metrics you have to add them point by point, but there is in general no way to decide which point to add to which point. So for a high-amplitude gravitational wave, we don't necessarily have a background metric, and without a background metric we don't have any way to define what we mean by the speed at which a wave propagates.
Andrew Steane wrote in a comment:
Rindler considers an extract treatment of a plane wave, and I expect it's in MTW and other well known books. Owing to the subtlety of non-static metrics it's fairly zany stuff but basically, yes, it travels at c.
I think the example you're talking about is in Rindler, Relativity: Special, General, and Cosmological, 2nd ed., ch. 13. Rindler shows that the wave propagates with coordinate velocity $dx/dt=c$, where the $x$ and $t$ coordinates are defined such that, in the flat region of spacetime that has not yet been visited by the wave, they are the standard Minkowski coordinates. He doesn't seem to explicitly justify the assumption that this coordinate velocity is correctly interpreted as the velocity of propagation, but given the setup, it's pretty plausible: observers in the region that hasn't yet been visited can synchronize clocks, etc. But this is just one example, and you can't prove a general rule from one example. MTW's example is a counterexample.
It would be interesting to know whether there are counterexamples where the velocity is equally unambiguous and is greater than $c$, rather than less than $c$ as in the MTW example. I would be surprised (and disturbed) if there were. Maybe there are general theorems that rule this out, but it's not obvious to me what is even the best formulation of such a conjecture.