Mineral spirits - wipe them with a rag dampened with it, then dry and buff with a clean soft cloth.

I just cleaned my oak cabinets. I used a product from the dollar store called Awesome Orange. Sprayed cabinets and used a damp terry cloth to wipe. Grease and oil came right off. Fantastic product.

Tks for the above product recommendations. I tried Bio Green Clean that is advertised on the radio and it did a decent job on the worst looking cabinet door. Took a bit of scrubbing though so I will try your recs for the rest of the cabinets. My last cabinets (Kitchen Maid) never left marks or they cleaned easier so I never had these smudge marks. These don't have as good of a finish on them, so I think that adds to the problem.

I JUST finished washing all the pantry, breakfast room & kitchen cabinets & woodwork, which had a lot of smudges around the handles. I used Mr. Clean Magic Eraser which does a much better job than a cleaner & washing cloth. The key to success with these erasers is to use them wet/damp - not dry.

There are generic versions of this cleaning eraser but I haven't tried them.

Here is a link that might be useful: Mr. Clean Magic Eraser

And here's a handy site for these erasers.

Here is a link that might be useful: 20 Unusual Uses for Eraser Cleaning Blocks