Ryan, welcome to Permies!
My first chicken coop was an old converted shed with a concrete floor. We kept it covered with straw and dry leaves as litter. The was especially helpful where manure accumulates, like under the roosting bars. The
carbon in the litter helps neutralize the nitrogen to keep down the smell, and by keeping it stirred up, composting would begin. I would toss down a handful of scratch in the coop every day and so the
chickens would help keep it stirred up. (Chickens adore scratching through things). When we cleaned out the coop, the dirty litter became the newest
compost pile.
The problem I had is what John mentions - except our were rats. They burrowed under the concrete and became a huge problem. We had something like a dozen baby chicks disappear one year, before we figured out rats were killing them.
When we finally built a new chicken coop, we kept a dirt floor. But we still use the deep litter method inside the coop, because we're pleased with how well this works.