We have St. Augustine grass and Bermuda. The free-range chickens spend their time pecking along the grass, but I think they're actually getting bugs and bits of seed or berries on the ground. The chickens that are in the chicken tractor have less of a choice, so when I move their tractor after ONE day, there is a rectangular patch of closely cropped grass. The interesting thing is that the chickens in the chicken trampoline are at least 8 different breeds, and they have different preferences. They don't get any greens unless I give it to them. When I mow the lawn, I toss in handfuls of fresh-cut grass clippings. Some go crazy for it. Others couldn't care less. Most of them love the immature leaves of the peppermint tree. Some like the peppermint berries or clover. Some like watermelon, and some don't! THAT was a surprise to me. I'd say that the ones that are most happy to get anything are the blue andalusions. They'll be the first to try something new.
My girls will eat grass, but what they really love is garlic greens. I had some garlic that had started growing, so i plunked it into the ground. I like to take scissors and cut them like chives, and put it in their cage for them. They love it more than anything else I give them.
They aren't into strawberries, though, which thoroughly confuses me. I went and bought a huge bag of garlic cloves, and plan on planting them on the entire outside of their run. So spoiled!