I hate to sound flip, but you won't have grass back there for long if there are chickens on it! It won't be long and that lush green carpet will be dirt, dirt, mud, and poop. I would suggest some kind of ground covering for the area in the run, whether it's pine shavings, true deep litter, or sand. With pine shavings and/or deep litter you can simply add a little more if it gets a little ripe back there. And a simple stirring of that litter will bury the poop and let the microbes and bugs in the soil break it down. With sand you can use a scoop similar to a cat litter scoop and pick up the messes, adding them to your compost pile. There are folks who prefer one over the other, so it's kind of your preference. But believe me, that grass will not remain in the run for much longer.
I prefer deep litter in my run. This spring was the first time it has had be cleaned in a year, and it really wasn't that bad. In fact, I left a good layer of the year old litter in there when I removed the rest of it, and added fresh material right on top. Easy, no odor, and compost ready for the garden!
There are plenty of threads dealing with deep litter and with sand. You might check both and see what will work best for your situation. But I don't think you'll be happy when the hot summer sun starts baking that accumulated chicken poo and it starts to smell. It's also not at all healthy for the chickens to be stomping, dust bathing, and standing in.
Whatever you chose to do, good luck with your new flock!