If you used smart strips, where there is a controller such as WS2801, WS2812, etc per RGB LED, then multiple strips would not be an issue. You blast out a stream of data to each strip, the WS28xx takes care of setting the PWM levels for its RGB LED, while they do that you send data to the next strip.
The strips use data & clock, or just data and derive a clock from the timing of the data line,
so you'll need to create a demux circuit as well to select the strip being commanded.
For example, run the clock (or data if there is only data) thru this chip
and control which strip gets controlled.
The unselected chips will see only a high output - if need the inactive level to be low instead, add 74HC04 to the outputs to invert them
6 inverters per chip, so use 3 if all 16 outputs are needed.
Libraries and WS28xx chips/strips can be found at adafruit.com and sparkfun.com.