Here is my 3x3x3 LED Cube project. It runs using a PIC16F690. Along with the brief explanation of how to make it, I have also included the .asm file for the PIC and a piece of software written in VB.NET that you can create your own LED patterns. These can be exported as .asm files and put into the main .asm code.
Step 1: Building
Each layer of 9 LEDS are connected with all of the cathodes together. The cube is multiplexed, meaning that only one layer is on at any one time. It happens so quickly that the whole cube looks like it is illuminated. RB5, RB6 and RB7 controll which layer is being illuminated.
From the picture you can see the connections to the top layer. I have only connected RC2 to show you how the connection should be made, the rest of the LEDs are connected the same way.
The easiest way to connect the LEDs together is to drill 9 holes into a piece of wood (picture 2), then place the LEDs into the holes, bend the legs and solder the cathodes together. After 3 of these are created, simple stack them and solder the anodes together. You should end up with 9 anodes and 3 common cathodes to connect to you circuit.
All you need to build this cube is:
1 x 5v regulator
1 x switch
1 x PIC16F690
3 x NPN tansistors
3 x 330 ohm resistros
9 x 220 ohm resistors
1 x 220uF capacitor
27 x LEDs
oh, and MPLab PIC programming software. (Other software may work, but my .asm file may have to be changed a little)