Help with cleaning linoleum floors?

I've got linoleum floors in my home and I'm having trouble getting them clean. I just use a mop with hot water and Pine Sol but the flooring has these little divets all thru it and they get disgustingly dirty. The only way I have found that I have been able to clean them is to use bleach and get down on my hands and knees and scrub the hell out of them. Obviously this is a lot of work and I'm worried about the effect of bleach on the floor, plus it's a pain in the neck using it and worrying about not getting it on my clothes. Can anybody suggest another method? I was wondering if something like CLR or TSP might work, but I don't know if they'd be too harsh on the flooring either. By the way, if you're going to suggest any cleaning products, I live in Canada, so some American products might not be available here.....