I wouldn't use hay, I'm pretty sure it molds easier than straw and other materials. One of the guys in our club used corn cob bedding in all of his breeding cages. He was a very good racer. I used it with my parakeets when I had them. The extra I had, I put in my pigeons' breeding cages and had no problem. They didn't seen to have any interest in the cobs.
Sand is a good bedding that is perfectly safe if eaten. It wouldn't be much different than grit. Turns the floor into a litter box. But you can only use the playbox sand that you'd use for kids.
Wood shavings and chips work too. Along with kitty litter.
I don't use bedding really. Occasionally (well, when I have it) I will put some shavings on the floor after cleaning. Once I was given enough straw bales to fill the back of our truck, so I used it for bedding.