You cant easily multiplex RGB leds as they all share a cathode.
Your best bet is the 595 I think. Its simple enough and cheap enough to make the extra complexity worth while.
The best way to do a 3d RGB box with them is to have one 595 per 8x8 grid sinking the cathode current and three (one per colour) sourcing current.
For a 8x8x8 box thats 32 595's for a grand total of $13.44 USD at Futurlec.
Depending on your application, thats a lot of LEDs to multiplex.
You may want to use two ATmega chips, one dedicated to multiplexing.