Years ago, I built a small cube, for a friend, as a gift. It was 2.5" or 3" to a side, and made from 1/8" white translucent acrylic. Common stuff. The light came from 4 superbright LEDs (red, yellow/amber, green, and blue). The super brite blues, and greens, just came out, so it had to be ~1995. Each LED was tied to a 4017 CMOS counter output. I looped around Q5 to the reset, so it would only sequence 4 steps before resetting. It was clocked off a microphone amp, but not straight from the audio, but from the envelope of its amplitude (diode and cap / demodulated).
They also make special RGB LEDs, now, that slowly sweep thru the rainbow, over a period, maybe as long as a minute. Check eBay.