When I had guinea pigs I had a C&C cage (cube grid wire & coroplast). The bottom for the cage was made out of coroplast that I cut to fit. It was pretty much a giant plastic, leak-proof pan with walls (perfect if you have carpet). The cage itself would sit outside of the pan. I also made a ramp, a 2nd level, & a hay rack out of the leftover supplies I had.
I lined the bottom of the pan with layers of towels then I put a large fleece blanket on top of the towels... that way the pee would go right through the fleece (you have to wash/dry the fleece a certain way for it to wick), making it stay dry. To keep the blanket on I would use binder clips to clip it to the sides of the pan. My g-pigs could not dig, pull, or get under the blanket this way, but I don't know if it would work with rabbits, as they are bigger & stronger.
I would wash/dry the towels & blankets 1-2 times a week along with "spot" cleaning every day.
After almost 2 years of constantly cleaning that cage I decided to sell my pigs because honestly, it was just way too much work. I loved them, but it was too much.
I don't know who poops & pees more, guinea pigs or rabbits! lol But considering the fact that you can litter train a rabbit, maybe it would be a lot less work than it is with the g-pigs. I've seen people use this same kind of cage for rabbits, except they either made it really tall or they closed it in.
Here's a pic of my C&C cage...
My Cage