Here's an idea I had when looking at the schematic for a Velleman LED dirver kit.
http://www.rapidonline.com/netalogue/specs/55-2036.pdfIt uses the same constant current driver but with three LEDs in series, each with a MOSFET in parallel so it can be bypassed to turn it off. A PWM circuit driven by three low frequency oscillators which will gradually shift causing the different colours to become out of phase creating different colours.
I probably won't build it as I have no need to a tacky mood lamp but I thought I'd post it anyway. The N-channel MOSFETs should be logic level so they turn on properly.