Thanks again for your helpful suggestions. I finished my cube just before Christmas, and to my amazement, when I first plugged the thing in it actually worked! I took photos as I went along, but I don't see any way to upload them on this forum.
For the benefit of others who may attempt this, here's a few notes from my building experience:
I found 16mm spacing was too wide for the LEDs I had. Some of the cathodes wouldn't touch adjoining parts with that spacing. 15mm was still a bit close, so I ended up using a 14mm grid. (cube is about 4.5") I used MS Publisher to print out a grid template with exact spacing, then temporarily glued it onto a piece of plastic for drilling. I started out using tape to hold the LEDs in alignment for soldering, but later devised a little peg system that I could move into the proceding hole in the assembly jig to align the LED perfectly. I cut a tiny slot in the peg with a razor saw and then added a notched sleeve from a plastic straw that held the cathode in place in the slot.
For the wiring on the back of the protoboard, I picked up the 31AWG wrapping wire you suggested (Radio Shack) and decided that it was just TOO fine to work with. After searching around a bit on the internet, I found some 26AWG sold wire on Amazon and ordered up several spools in different colors. I found I could fit 3 wires into a hole on the protoboard and the wires was still stiff enough to bend them into the shape I wanted. I still ran into problems when trying to blob solder between the D lines where 3 of them connect. I ended up having to use a solder sucker to clear out the mess I had made and then just daisy-chained the wires from the other end of the ICs at the outer edge of the board. I was about ready to smash the whole thing.
I looked for a long time to find a suitable enclosure box that I didn't have to build. I was looking for something in plastic, but ended up with a wooden "photo storage box" from a Michael's craft store. I dumped the cover, cut out a 3/4" plywood base that fit internally and and made the bottom of the photo box the top of my enclosure. Added a couple coats of poly to smooth out the wood, then topped it with flat black paint. Used the same grid template to drill holes for the anodes in the box. (measure twice to be sure it's centered exactly, I was p**ed when I discovered that mine was 1/4" off center AFTER I had it drilled) Doing it again, I'd make a removable TOP instead of a bottom. It was a bear getting the wires soldered on the anodes, so much that I ended up crimping pin sockets onto the wires and then soldering them on the LED leads. Cutting the ribbon cables also turned out to be a challenge. They didn't seem to want to tear, so I ended up cutting strips of 2 wires about 5" back with an x-acto knife. I just left the unused wire attached to support the othere conductor. Even with careful guidance, I had a couple spots where I cut through to the wires and it was a real pain to repair and re-insulate them.
Now that I'm done with construction, I've discovered that I have definitely outbuilt my programming experience. Maybe I should have started with a 2x2x2
I'm still going through the code, trying to figure out how I can program some of my own effects. I started with Arduino last winter and got up to speed pretty quickly on the basics was to the point to where I was writing my own sketches using multiple shift registers. I had to drop the hobby over the summer due to work issues. It's definitely going to be a (re) learning experience. Then again, that's why I took it on. Thanks once more for all the help.