If you throw straw in you really don't have to take it out. The chickens will have a ball scratching in it which will get air mixed in and it will dry up quicker. They will also break the straw down turning it into compost. My run basically acts as a compost. I dug dirt out of it for the garden this spring. Dark and rich.
Some will put sand or a drainage system of some sort as mentioned.
Another option is a partial roof of some kind.
It probably depends on your situation and your area. Mine for instance I use straw, grass, leaves, garden, some sand, and yard debris. Why? All are readily available since I live in a rural area. Heck I can just grab the wheel barrow and go get what I need. This reminds me I need to go to the river to get some sand(have to drive to it though). Now if I was in the city throwing leaves, yard clippings, and things considered waste might upset a neighbor since it may look "trashy" or messy. So maybe straw or shavings so it is more manicured looking may be the ticket. Just add as it is needed. If it gets really nasty or smelly then scoop it out to compost further or till it in in the run. I till mine at least twice a year and scoop it out in the spring for the garden.