I suggest tpic6b595 chips. They are like a 74hc595 shift register but with higher voltage and current capabilities. Each chip has 8 outputs which have built-in mosfets and can drive your groups of LEDs. Two chips, daisy-chained together will give you 16 channels and needs only 3 Arduino pins.
Tpic6b595 outputs can switch up to 150mA and up to 50V. If you can find them more cheaply, there is also tpic6c595 which can switch 100mA and up to 33V.