LED Multiplex Anode-Row Driver
Have you looked at Maxim's range of LED drivers? For example, the MAX7219 can drive an 8x8 array of LEDs, so you'll just need four of these to drive your 16x16 array. They're intended to drive 7-segment displays, but they can just as easily deal with any LED array.



A completely different solution you might like: RGB LEDs with integrated WS2811 IC.



Each LED is serially addressable. You just chain them with each LED's serial in connected to the previous LED's serial out. Then clock all your data into the first LED. This makes wiring up a display pretty easy. They're only $0.145 each from Ray Wu's store in China.