[thisshitisannoying]

There's a lot of ways you could do this. The easiest thing I can think of is to just get a normally closed switch, put it up against the cabinet door, when you open the cabinet door the switch closes and electricity flows to the LEDs. I did this before in an old vehicle that had no light in the glove box. Keep in mind you'll be able to see the switches so if it's for something you want to look really nice maybe stick with the sensors.

[thisshitisannoying]

A simple search on ebay for "mini pir motion sensor" returns a lot of good results. You can find them for less than $3.
You will have 3 pins on the sensor, Voltage In, Output, and Ground.
Parallax also sells a mini PIR module for $9.99 they have good quality products, you get what you pay for.