Simplest circuit that flashes and fades color LEDs synchronized to music?
Depends on your meaning of simple. @Tcrosely answer gives a good example, a simple passive color organ, all parts that can be found through hole. Using resistor/capacitor pairs as bandpass filters.

Or you might think a microcontroller solution is simpler. Which there are three ways of doing.

Completely in software (FFT libraries for Arduinos/MSP430/Pics), using another chip to capture the audio and turn it into digital information you can use (The MSGEQ7 + microcontroller is a popular version of this), or dedicated chips (All do three band, bass/mid/treble, with adjustable cross over points, or randomization of the LEDs)

TI's/NatSemi's LM4970 (most ""hobbyiest"" ""friendly"" package) or other's in their Boomer family (really small SMD). The LM4970 needs to only be enabled by i2c from a microcontroller, while some of the Boomer chips have an auto on mode.
AMS's AS3665 and AS3668, both which have auto audio sync mode.
ISSI's IS31FL3193,IS31FL3196, and IS31FL3199. Same as the LM4970, needs a simple i2c on command.

Simple really depends on your experience and how easy you can get parts. A dedicated chip method would need a lot less parts or board space than a passive component color organ, and can be powered by a lower voltage.