I use deep litter (mostly leaf/stick gathered from the trees surrounding the cleared portion of the pasture) and what straw is either kicked from the box or made from ripping out/cutting down tall weeds in the pasture, on top of dirt. Well, technically, its clay sands and sandy clays - dirt would drain better.
One of the first things I did, after fencing the run, was carefully observe rainwater runoff following a heavy storm, and regrade portions of it to take advantage of natural tendencies in the terrain.