I use the deep litter method in my brooder, and I fully intend to use it in my duck house as well. Peat moss on the bottom for absorption, about 3 inches thick. Then a few handfuls of garden dirt on top to provided starter microbes. Next a 2-3 inch layer of pine shavings (less absorbent than peat, more absorbent than straw) to help with absorption, and to provide a barrier to keep the ducks from dabbing down into the peat. Finally, a layer of straw, about an inch thick. I add more straw as needed. Once a week I go in with my garden claw and mix things up a bit. So far, it's working well.
Hay is not a good idea for a couple of reasons. First, it produces mold and mildew much too quickly, and some molds (aspergillis comes to mind) are toxic to poultry and other small animals. Second, most people intend to use the duck litter in their gardens, and hay is full of weed seeds. So unless you have a plan in place to hot compost your litter, or you have an endless amount of time for weeding, you should use chopped straw.