Tonight I went to Home Desopt and got a scrub brush, sort of like this one, with about a four inch square head and a short handle. I also got some quarter-inch bolts and nuts.
My design is less involved than the one Elf linked to. I hacksawed the handle off of the brush, then drilled a hole of just the right size through approximately the center of the brush head. I screwed the bolt through from the bristle side, and added a nut to keep it in place. I don't care if the bolt head pokes through the bristles, since they're nearly two inches long and fairly thick-- I just have to be careful not to press too hard.
Tonight I did a quick trial run that involved one small tiled section of wall. I dipped the bristles in water and Comet, and attacked the wall with the brush. The grunge came right off, though it was a relatively easy case. Soon I'll do an actual trial with actual grotty tile and see what happens.