As someone else mentioned on another thread, you might try Goo Gone. Or fingernail polish remover. But I don't know what that would do to the finish of your cabinets.

I would try cooking oil. Corn oil or the like. It won't hurt the finishes, and fairly often oil will remove the sticky stuff left behind when I soak a label off a jar. After it soaks a bit, try scrubbing with a plastic scrubber.

Then, of course, wash the residue off with soapy water and dry immediately.

If you find something that works, do post the results for us.

By the way, I use Cushy Cupboards and love it on open shelves and in some drawers. Much better than other shelf lining I have used. Much easier to clean than the ridged stuff.

Careful with that Goo Gone! I just ruined something last week with that stuff because I failed to read the directions carefully. It can definately damage surfaces, so use with caution. Maybe you can try heating up the shelf liner with a hair dryer ... that works amazingly well to remove all kinds of sticky stuff, so it's worth a try.