How can I remove stickiness from 30 y/o cherrywood cabinets/drawers?

What is the stickiness from> grease? oil? You can try TSP or Krud Kutter. Make sure to test it in a small area before putting it everywhere.

I would try to use lemon essential oil

This may give you some ideas: https://speedcleaning.com/2016/cleaning-tips-from-the-clean-team/3-easy-steps-remove-sticky-buildup-wood/

Hope this Helps! Jeremy - https://do-daddy.com

Clean with a good degreaser/cleaner. I haven't found anything that Awesome cleaner from the dollar store doesn't clean.

If the wood is finished, use murphy's oil soap. It can remove 40 years of grimy buildup, quite rapidly, using some elbow grease and a clean lint free cloth. Rinse lightly and buff dry.

It really depends on what the sticky stuff is. In an inconspicuous spot, try TSP substitute in a spray bottle mixed with water and a rag. Cleans grease right off with zero effort.