Straw.

I throw the straw in the cages every so often and due to the wire floors and
the rabbits moving it about it just makes a fairly even layer on top of the
manure. Once the area beneath the cages if filled, I fork it out and toss
more straw down on the bottom. Usually by the time I clean out from
beneath the cages, the bottom layers of straw and manure have started to
compost. When I hit the spot were all the rabbits pee, it can be a little
smelly, but, once it is mixed in with the rest of the manure straw mix it's
not noticeable unless your right on top of it.

Good straw works wonders for absorbing the excess, and between the worms and
fungi, break down fairly quick. Once it is broken down, it gives the soil
good texture.

It should be some what fluffy, and not just a collection of course stems.
The course stems do not do much good except to put down over mud so as not
to track it every where.