When you multiplex, the LED is not on all the time --> it appears dimmer. The more LEDs that have to share their turn for being on, the dimmer it gets. Fact of life/multiplexing.
You can partially counter this effect by using "too much current" when it is on, but you wear the LED faster, you will damage it if you occasionally leave it on (f.ex a program flaw). Not recemended.
On a 4x4x4 your mutiplexning should be have them at a 4th of their "brightest" value, if you multiplex right. Your code seem to be lighting them a 16th of the time. Rethink your wiring/multiplexing.
Others use extra chips to control the LEDs for the bigger cubes to overcome the IO pin limitation. - TLC5940 is a good example.