Most people seem to agree that Magic Erasers with alcohol are the most effective method for removing ball swirl marks, but there's some dispute over the consequences. I've been doing some tests and I thought some people might find this information helpful.
First, here are before and after pictures of the area I'm working on. I would call these swirl marks medium density. The playfield here is Taxi, so it's System 11, with no diamond plate. I cleaned it with a 1" magic eraser cube and 91% alcohol. I used almost no pressure, and moved in circles and various directions until I felt the results were good.