Those microfiber cloths...Scotch makes them...they are washable and reusable. I buy a pack of 25 at Sam's for about $10. They are WONDERFUL for cleaning. I would make up a good strong cleaning solution (like water with pinesol or water with Dawn dish detergent) and try washing with that. Dawn cuts grease really well and the microfiber cloth has little nylon loops that act like little blades to scrape off built-up stuff. They are WONDERFUL...
Another thing- if it's really thick, you could try one of those scrapers from Pampered chef for cleaning off crud on the stone bakeware. It looks sort of like a plastic disk...try softening it with dawn solution and then see if scraping with the plastic scraper helps.
It's probably going to be a several step process. Don't get disheartened...it will get better.
If the paint surface is really solid, you might try those green scrubbies (or yellow or white scrubbies are not as harsh) and see if mildly rubbing with that helps.
Good Luck!