With a 4 channel 12V relay module (e.g.
http://www.lightinthebox.com/4-lines...prm=1.2.1.0 ), and an RGB strip of LEDs (e.g.
https://www.sparkfun.com/products/12023), it should be straighforward. You would use three DIO ports, one each for red, green, and blue, to control three of the relays, each of which controls power to one of the colors. For this application, you want a "non-addressable" LED strip. This means that all of the LEDs of a given color all go on or off at the same time. Addressable strips would require a microprocessor or a fair chunk of CPU time to send specific colors to specific LEDs on a strip.
Edit: fixed first link. Also, note that I recommend NOT using the ground jumper for a 12V relay module - let the opto-isolator do its job!