Hi G. - the key is to remove the sticky grease without ruining your oak cabinets - the steam cleaner idea might raise the grain and bleach the wood. Chemical degreasers could also remove the finish. Whatever you try, do a small patch test first.
I like to use Murphy's oil soap on my oak cabinets. BonaX works great on my hardwood floors, it should work on your cabinets too. I've also heard of someone using straight Spray-n-Wash or Shout as a degreaser, but again do a patch test on anything you know isnt made for wood. Have you tried just using some Dawn dishwashing soap with water yet?
Guardsman also makes good furniture cleaning and polishing products.
My trick for getting grease off of my gas range has always been a heavy dose of turmeric and a little water and allowing the paste to soak and then scrubbing off but this could be too abrasive for your cabinets.