Most people use straw, but when I sow grass at home, I use hay. (Always have plenty around!)
It's usually cheaper so you can spread it thicker for less money.
After all, the Idea is to keep the seed and ground moist. Straw is coarse and lets moisture escape quicker.
Hay laid more dense seems to work better for me, just don't get it too thick so as to trap the moisture bad enough to make the seed rot.