Get a piece of scrap wood and drill 16 evenly spaced holes into it. The spacing depends on the length of your LED leads. Drill the holes big enough to fit the LED into it. I used a piece of pegboard.
Step 2: Solder all negatives together
Bend over the negative lead so it is touching the negative lead of the next LED beside it. Solder all negatives together in one layer but be sure not to bend the positive leads over or get solder on them. Repeat this process 3 more times so you end up with 4 seperate layers.
Step 3: Solder all positives together
Now that you have 4 seperate layers, take one of them and insert it back into the template. Now with the positive leads sticking straight up, get another layer and place it on top of the one in the template. Connect the two together by soldering the positives together in a column. Do not solder anything to the negatives you have already soldered together. Repeat this process for the remainder of the layers until you have the cube built.
Step 4: Wiring
Now theres a lot of wires to connect so make sure you put every one in the right spot. Insert your newly constructed cube into your protoboard and solder the positive leads from the columns in. Beside each lead that you just soldered in, insert a 100 ohm resistor and solder one end of it to the lead. The other end of the resistor has a wire connected to it that will later connect to your arduino. There will end up being 16 of these. Now you will need to solder a wire to the middle of any negative in the top row of your cube and do this for all four of the layers. There will be 4 of these.