I've tried a few different cleaners but nothing seems to be able to remove the scuff marks.
Marc, we sympathize. We have tried to find out what causes these marks, but with no luck. We've had better luck with removing scratches like this, though. Bon Ami ($1.44, Amazon) is great for this. Also try Barkeeper's Friend ($7.99, Amazon). This powdered cleaner is rougher, so try it on a plate back first, but it can take off deeper scratches. It's designed for use on porcelain sinks so it should do the trick on plates as well.
Anyone have a good answer for Marc on how to prevent scratches on his dinnerware, or get rid of the ones he has?