I'm afraid I have very bad news 
A few days ago I've started working on a 4x4x4 cube to see how I can build it, before I jump into the real one (the 8x8x8 cube). So I drilled a 4x4 matrix of 3 mm holes into a piece of wood to help me assemble the horizontal layers of 4x4 LEDs, filled the holes with 3 mm LEDs, cut the craft wire to pieces of appropriate length and set out to solder the pieces of craft wire to the LED anodes. Now this already turned out to be a huge pain. The craft wire is 0.8 mm thick and the LED legs are way thinner so it's close to impossible to hold them together with clips while you solder them together because the clips will only close to the thickness of the craft wire and will not be able to grab the LED legs properly. Because of this the building process was about 3-4 times slower than I thought it would be. I have wasted hours and hours trying to hold the LED legs together with the craft wire, while I was soldering them. Anyway, after about 8 hours of work I managed to make 4 layers of 4x4 LEDs connected by their anodes into planes.
While I have built the layers, 2 LEDs burned out or something, because after testing the layers, they were not working anymore, even though they had worked before being soldered into the layers. I didn't think much of it at the time, I just replaced them with new LEDs. But the ordeal began when I tried soldering the cathodes together into columns. Holding the LED legs together with the craft wire was a pain, as usual, but that was the least of problems. After soldering the cathodes of 3 layers together (into 16 columns), several LEDs started not working and instead they seemed to conduct electricity directly, without lighting up, which resulted in several LEDs lighting up instead of just one when I tested the intersections of layers and columns. In other words, when I put the multimeter's positive lead to layer x and the negative lead to column y, instead of having only one LED light up at (x,y), I saw 2 or 3 LEDs light up. This was not for all (x,y) pairs, just for a few. I managed to find some LEDs which did not light up at all, so I figured that they are the problem. I managed to solder them out and solder new ones in their place through very painful work. Imagine if this would have happened in the middle of a 8x8x8 cube, it would have been impossible to replace them. But the problems were far from over. While I replaced those LEDs, others have burned out. And so on, until I had to throw out the whole darn thing into the garbage. Hours of work gone.
Now the question is: what made the LEDs burn out in massive amounts?
- My best guess is that they are burning out because of the heat from soldering.
- My second guess is that perhaps the multimeter's diode testing mode is burning them out.
- Is there any other explanation?
God knows I've tried to limit the soldering time to 3-5 seconds, with a few exceptions. I can't do it faster, the solder just won't adhere any faster. Maybe the soldering iron is too hot? I think it's a bit above 300 degrees Celsius, although it doesn't show the temperature, so I'm only guessing. Turning it lower seemed not to heat the solder fast enough and I had to keep the iron on the LED legs longer because of this, so it wasn't good either. I've watched several videos about how to solder properly and I'm using standard 40-60 rosin core flux, so I don't think the problem lies there.
This is such a disappointment. After spending 150-200 $ on components and who knows how much time on planning and soldering, I'm close to giving up the whole project because I just don't see how I could avoid these problems. This wast just a 4x4x4 test cube. If just one single LED fails in the middle of a 8x8x8 cube, I'm screwed and my whole work goes down the drain, so I'd rather not even start working on it. The only solution I can think of is to somehow solder the LEDs together with some non-traditional, cold soldering method. Honestly, I've never heard of such a thing, but hopefully it exists, otherwise the whole project is wasted.
Maybe the LEDs that I've ordered are poor quality and weak, maybe they can't take as much heat as regular LEDs? Or maybe I should try soldering at really low temperatures? I doubt that it would work. Maybe I'm burning them out with the multimeter's diode test mode? I don't know. It all seems really hopeless right now...