There isn't an answer to this one. There are some strategies for doing this with restrained box sizes but the general case you describe isn't actually solvable. At least no one has yet. (of course, there may be some genius out there that just broke NP-complete problems too but...)

I've not heard of anyone solving this or any related 2d or 3d problem.

The balance problem, aka tape-sides is a good simpler example. Given a set of weights (or a set of song lengths) divide them into two sets that have the least difference in weight (length)

There are a lot of nice strategies but they all have nasty cases that don't work. Exaustively trying every case is the only hope. The best bet is to minimize the cases you have to exhaust. Force everything to inch(cm) boundries, move boxes on the same. Pack things around the largest volume item, etc.

Really, not to discourage you too much, it might be a good idea to start on something a bit simpler, like curing cancer. =)

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