I've always used Clorox (which is basically bleach) or diluted bleach to clean my tubs, bowls, hides and have never had any problems. I spray it down, wipe it with a wet rag to make sure that all the surfaces are covered, rinse REALLY well, and then either let it air dry or dry it with a clean towel.

I just got a 2' x 3' cage and obviously I'd have problems rinsing it out, so I bought chlorhexadine to use in it, but I still use clorox on the hides and water bowl.