Well I couldnt wait and modified the code to SPI.transfer. Still no problems with ghosting...
link:
https://www.dropbox.com/sh/5jzjoelre7b6h4g/xAy4sVeAhI/LED_CUBE.ino
Edit: Still the code I'm using is not a good way to program the cube. After a while u will run out of memory and the cube is going to do strange things. Even now I'm using PROGMEM its not working. Is it possible that you can post your code as well??