Here's another example
http://www.jameco.com/webapp/wcs/stores/servlet/Product_10001_10001_2169539_-1
Read the parts list - includes Uno R3, 8 shift registers, and one 3 to 8 decoder (which can be replaced with another shift register)
This one uses 74HC574 to source current to the columns, and NPN transistors to sink current from each layer.
As you can see the 74HC574 is not rated to provide 20mA for 8 outputs, so brightness will be limited.
Using PNP or P-channel MOSFETs to source current to each layer and TPIC6B595 to sink current from each colum will allow full 20mA per LED if desired.
You can use PWM on the shift registers OE/ pin as well to provide 0-255 levels of brightness control also.