You'd want a LOT of material on top if you used landscape fabric -- chickens can dig dusting-holes pretty deep, like 6-8" or more, and you sure would not want them exposing landscape fabric (which they would scratch up and then eat, and it's not like the world's most digestible material). I...
Sand or gravel work great to avoid mud... but only if you put them down BEFORE you have mud (or once it has thoroughly dried, in the summer). As I've said elsewhere a lot of times, if you put sand or gravel into mud, the mud just swallows 'em up. You don't even get sandy or gravelly mud. It is...