Even if you go with a mega I think RAM will be a problem. 24x24x24 x 2 LEDs will need 27K of memory at 8 bit resolution; you'll need even more to go to 12 bit.
As Grumpy Mike says I also think you'll have to drive multiple planes at a time. 24 planes x 256 slots of PWM x 50 Hz (so the eye won't see the redraws) is about 300 000 updates per second. That only gives a 16 MHz processor about 52 clock cycles to get ready to update the next row, which I don't think is possible.
Good luck!