A friend showed me these days a really nice piece of software called Sprint Layout, which can be used to design custom printed circuits. I started playing with it and found that it's very easy and straightforward to use. Driven by the desire to have the whole LED cube driver circuit on a single board, I've taken CrossRoads' design and tried to create a drawing for a printed circuit. Here's how it turned out:
Back:
Imaginary Front:
Real Front:
The board is 180 mm x 120 mm and holds all the components (except the Arduino and the LEDs, of course). The connections to the Arduino are marked with large rectangles and the connections to the LED cube are marked with small squares. The round connections are all internal.
I've struggled with it for about 6 hours until I have reached the current result. This is the first time in my life that I've ever done something like this, so it might be far from perfect. It might even have errors in it, unfortunately it's 3 o'clock in the morning here already, so I have to go to sleep, I can't verify it... If you guys notice anything strange or you have suggestions related to how it could be improved, let me know 
I've spent quite a lot of time trying to squeeze things into the smallest space possible. A multi-layer PCB would have probably been smaller, but I don't know how to do that and I didn't even want to work on more than one layer because it might complicate the creation of the board.
Now, I haven't got the slightest idea yet if and how I can turn this design into a real board. I don't know anybody or any company who will accept such a design and make a board out of it. For now, it's just a design. If I can turn it into something concrete that's going to be great. If not... well, it has been a nice learning experience