Building your own RGB LED cube is an interesting challenge, apart from the physical work required to carefully assemble your own the concepts with controlling the LEDs is also an interesting learning experience thanks to the concept of "charlieplexing" - controlling many LEDs from a few digital output pins on your microcontroller.
A great example of a home-made Arduino-powered RGB LED cube has been documented by Instructables member HippyNerd which results in a neat cube that instead of using PCBs for each later, relies on a very carefully-arranged network of bare single-core wire for rigidity and connection.
Apart from building the cube there's also some interesting explanation about how the circuit works. Finally, a demostration of the cube is shown in the following video:
If you're interested in RGB LED cubes - but not up for making one from scratch - check out our CUBE4: RGB LED cube kit:
It's easy to construct and use, and with an onboard Arduino-compatible board the cube can be used to display all sorts of data or create visual effects. It's easily controlled via simple text commands via the USB port - or write your own Arduino sketch. It's incredibly customisable and there's so much more. For more information and to order, visit the CUBE4 page.