So, you will have 24 planes of 8 LEDs x 8 LEDs, with 3 colors per plane co-located.
Start simple - in plane 1, you will have an 8-bit shift register in the X direction (left-right), and another in Y (back to front).
To turn an LED on, you will drive its anode high from one shift register, and its cathode low from another.
To go up (Z direction), you will have another shift register, its output will enable the X-Y shift registers on a plane by plane basis.
Be careful building it, the fun part will be the software