All you need is some thin wire, a contact design to complete a circuit when the door is closed and a single LED driver you can mount on top of or in one of the cabinets and you are good to go. The driver you need would depend on the number of the LED's you want to run, but they are such low power consumption that you would have to have a huge set of cabinets and pulls to need a very large one. And just think, I am sure the whole thing would be considered to be "green" with the minimal amount of power the lighting would consume.....
Typical LED forward voltage is about 2V (more for blue & white); all you need to worry about is current limiting, so if you just series enough LEDs to match the supply voltage then that's it, no driver, no extra resistors, no nothing.
My dresser has 4 knobs per drawer (which used to be two pulls per drawer), so I could make up a little brute force 8V power supply and run 4 2V LEDs per drawer with no extra circuitry; I'd just need to supply enough current to drive all drawers at once, which still doesn't add up to much-- definitely "green" accent lighting. Bet nobody else has lit knobs on their dresser...
Some little variable wall warts produce 9V, that would be 2.25V per LED which is probably fine; might not need to make my own power supply at all. This is definitely doable. It'd be nicer with a photocell to only light them up at night tho.