I've got those painted gray 15" rims with the black plastic center cap, and a couple of them are stained pretty bad with brown grime. I've tried polishing compound and metal polish but neither took more than a tiny bit off. Do I need to strip the paint off?

I haven't tried this, but I wonder if a claybar will remove the brake dust grime. You might want to give it a try.

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87 IROC 350 TPI with less than 10k original miles
93 Z28 Pace Car #494 of 645
My IROC Website
Northstar Camaro Club of Minnesota

I had the same problem when I bought my car and eventually decided that the only thing I could do was to repaint them. I bought some spray paint remover from wal-mart and started taking the paint off and the first application removed all of the grime and they looked brand new. If you try this it would be a good idea to put the paint remover on a paper towel and wipe it onto the wheel and immediately wipe it off with a clean towel so it doesn't take too much off. Maybe try testing it on a small area first to make sure it doesn't take too much off. Hope this helps.