I move the guinea pig run across our lawn. Once they've cut it down "enough" I move it, so that it overlaps slightly. The wild bunnies do help. It's funny when they look at each other. Cavies inside the run, bunny outside, all of them chewing.
With chickens inside, this would be called a "chicken tractor."
We have chickens, but they free-range for the most part, until they're locked out of the garden. They don't chew the grass, but do eat lots of other things. (And some plants, hence being locked out the garden for summer.)
We have very few rabbits around me. I've heard it's because the nearby golf course provides space for foxes. Otherwise I imagine they'd struggle with all the backyard fences.
My real goal is to provide a maximally inhospitable environment for cockroaches of all sizes.