OK, all problems are now fixed. After soldering in all the resistors for the anode layers and cathode columns, the ghosting is completely gone with 30 microseconds software delay after turning everything off before turning a new layer on (during multiplexing). I've updated the schematic and the printed circuit design accordingly:
Unfortunately the printed circuit has now grown to 14 x 12 cm because I had to fit 72 more resistors on it. I've already ordered one from a guy in town who makes PCBs, it cost me about 17$, let's hope it's perfect 
As I wrote earlier, these resistors increase the power consumption of the cube by 72 mA in passive mode (when everything is off), which is something that I'm not happy with at all, but it's still better than ghosting...