It's a little difficult to say that it will be a definite fix but a great start would be refinishing the inside of the cabinet. While I'm sure its possible to re-stain and seal the inside of a cabinet that just seems like way too much work so I'd stick with painting.
I found this nice article about painting cabinets if you want some design ideas. The process shouldn't be all that involved. You'll probably want to sand first and clean the shavings out well then give it a nice coat of whatever color you decide to go with and let it dry. Pretty much like painting any other piece of furniture, just a little bit more of a pain to get inside there.
I once stripped and painted all the drawers and cabinets on a work truck and that was a real pain. Luckily I was able to use a paint sprayer so that made it a heck of a lot easier. If you know someone with a paint spraying system beg them to use it and the finish and coverage in the tight areas will be wonderful.
Just my $0.02 and I can't say that this will be a definite solution but I can't imagine a wood smell getting through a through coating of paint. If your really worried you could use some thick outdoor paint that's designed to work as a sealer too. Best of luck with your project, post pictures if your need help along the way!