Ladies and gentleman, it's alive! It works! 100%!
I've even tried multiplexing and that also works well. At 4 ms multiplexing cycles it looks nice, at 6 ms my eye can already see the "trick" (it starts vibrating). I'll probably aim for 4 ms or lower.
I would like to thank you all for all the help you've given me with this project. I wouldn't have done it in a thousand years by myself, without your answers. My thanks are aimed especially at you, CrossRoads! Your design and your help is something that MasterCard cannot buy (priceless!). Seriously, I cannot thank you enough for having the patience to assist me across the steps of building the cube. If there's any way ever I can hep you back with anything, don't hesitate to let me know!
Now that the 4x4x4 cube works, I'll continue experimenting with writing software for it. After I'll have something usable, I will start putting together a 8x8x8 cube I'll start by turning that printed circuit design into something real and transferring the current 4x4x4 cube onto that board, populating only part of it.
@Hippynerd: It's very late at night again here, so I'm going to sleep now, but if you're still interested in that BOM, let me know and I'll gather my notes and write you a comlete list with stuff that you need for the cube and the approximate prices for which I've bought the components.
The 4x4x4 cube is just a test cube, but it's a great success for me. From here to the 8x8x8 cube it's just an immense amount of soldering. The knowledge is in my head and it's all tested now. This is a great day for me I've wanted to build a LED cube for 2-3 years now, but I didn't have the necessary knowledge. I've learned a lot from this 4x4x4 test cube. Now I can build the real thing. This is a dream come true.
Thank you all for everything again! I am most grateful for your help.
I'll borrow a camera soon (mine got broken) and I'll be back with a demo video of the 4x4x4 cube in a few days! Also, I make it my goal to thoroughly document the process of building the big cube. I will write a series of articles on my blog about it, which will help other people achieve the same result.
Hooray!