Boiling water and baking soda will bring the dirt to the surface, and you should be able to scrub it away with the magic eraser. It usually takes a few tries to get it all off. Or, if you are impatient like I am, let the dirt come to the surface, and then use a cotton swab and acetone, it should get it right out.
I don't care what anyone else suggests, for the love of ponies don't use oxy clean. You are setting yourself up for disappointment. Use BAKING SODA!