How would sheep and rabbits work then? And you could make a nice game pie out at the end too.
This can be done with chickens as well. Google ‘chicken tractor’ – but you’ve got to move it everyday, and there is the fertilizer.
-rainy
James Herriot the famous English vet/author told about the time he was called to a retching goat. He put his finger way back in the mouth of the goat, and found an elastic band. He pulled out the farmer’s longjohns, which had caught on the tongue of the goat. “That’s where they went!”, cried the wife.
In my experience with backyard chickens they don’t eat the grass as evenly as, say, guinea pigs. They seem to be better at scratching up the ground and eating all the greenery, leaving bare dirt behind (perfect for planting).
Welcome to the boards, ellere!
The answer to the question you seek goes beyond you. Our limitation is suburbia. What got me thinking the exact thing you are thinking is reading a book called Omnivore’s Dilemma by Michael Pollan. In the book he interviews a man who has figured out how to run a farm without all the chemicals. He calls it grass farming. No animals on his farm are in static pens. He has a field, it has grass. Once the grass is a certain height, he moves the mobile chicken pen over it, few days later, he moves the cows over it. Then the cycle starts over. Chickens rough up the ground, eat the bugs and such and process the leftover cow poop back into the ground, they poop, fertilize the grass some more, the cows come in, eat all the grass, viola! Beef.
As far as suburbanly acceptable grazers, the guinea pigs appear to be your best bet, the question is, would they do an entire area effectively, or would you spend more time/energy, moving their mobile pen?
City folk are so squicked out by animal poo I can’t fathom them seeing anything other than a standard dog or cat as glorified poo factories. I personally would love a goat, heck, I’m considering renting one. Goat poo has to be the tidiest poo ever. I’m not sure if they have come out with any mini sheep, but I would bet you would have hell to pay with the neighbors.
BTW, never count on a rooster to wake you up, they crow all hours of the day and night, don’t believe everything Bugs Bunny told you.
Also, one of the BEST alarm systems you could ever have is a herd of geese. It takes a string about 6 inches of the ground to keep them in and they HATE intruders.
Excuse me while I wander off to the woods with my goats and geese. So I can live how I WANT to live.
Fishing for a link? 
The Scrivener, have you been reading Jean de Florette?
Just don’t start digging a well on your property.
Hijack alert - Since we’ve a goose person to ask - Hijack alert.
I’ve been trying to find a way to keep the grass down in a grove(?) of blueberry bushes where the limbs droop so under the weight of the berries that mowing becomes nearly impossible. I once saw on television a Hawaiian coffee farm that kept geese in the grove(?) to eat the grass down. I surmissed that the geese disliked the coffee fruits and left them alone. I fear that geese might find blueberries tasty and eat them instead of the grass. Thoughts?
-rainy
This is absolutely brilliant.
Now I really want one of those German giant rabbits as yard animal, to keep the stupid wandering neighborhood cats out of the yard.
You guys are posting some weird links… and re. reading Jean de Florette, no, but I saw the movie after it came out. I’m afraid it doesn’t necessarily pay to be too enterprising and inventive no matter where you may live.
And to hijack a hijack, my WAG is that rainy’s geese will be only too happy to eat the blueberries… leaving behind “gooseberries” in return! 
I have run this proposal across my financial people and they have advised me that it would be more cost effective to employ a gardener for the next 37 years.
You now owe me $23,419 in consultancy fees.
Be sure Hal Briston doesn’t get near the lawn if you do decide on sheep…
We’re not even close to done with making fun of Hal, right?
Given that sheep are herd animals, would it be humane to keep just one all by itself in your backyard? Would it be frightened or lonely without companionship, or would it be perfectly happy to graze alone?
ha! Thanks for the laugh.
Re: goats
To hell with the grass, they might be worth it just for their entertainment value. When I was a kid, our neighbors had goats. They are quite brilliant problem solvers. I once saw them cooperating in the following fashion: One goat (goat A) was laying on the ground near a fence while another (goat B) stood on A’s back with its (goat B’s) front feet on the top wire of a cattle fence to reach the tasty tree-leaves above them. Also, they have amusing little nose-lips and will lick you and try to eat your hair. Yes, they will desecrate your foliage, but isn’t it worth it?
Artificial lawns have come a very long way in recent years. This isn’t your father’s astroturf.
Oh, yeah. A dozen chickens can turn 150 square feet of grass into bare dirt in under a week.
You mean I wasn’t supposed to turn left at Albaquoikey? 
We pen up the chickens (and rooster) at night. He doesn’t make noise until we let him out in the morning. Not much noise, anyway.
We’ve always kept our hooved mammals in groups of two or more. Sheep and goats interact and play with each other, and as they get older, they just keep each other company. They’re definitely herd animals. Get two, not one.