I have done this in my run. I mowed my front lawn (I do not use chemicals or fertilizers) and kept 3 of the mower bags for the run - just dumped and spread it all out. I've also piled in a bunch of straw. I go in and rake it over sifting it around each morning, and the chickens are also scratching it and turning it all day. So far, no smell and even with some good rain, the top layer stayed very dry (I have a roof on my run). I plan on raking up all the leaves that have fallen into my front yard this weekend and adding them to the run too. In just 4 weeks I can tell that it is composting and breaking down into the ground. Providing a good compost-able cover to the ground benefits the soil, which in turns brings in the worms/bugs etc. that the chickens find for a little snack. They have really fought over slugs and grub worms.
Plus if you have access to clean grass and leaves all year, it's FREE!