The best solution is to neatly break down most of the boxes flat (whenever possible), but save a couple bigger ones to put all the Styrofoam, plastic, etc. into.
Use a marker or something to mark which styrofoam, etc., belonged to which item if you can't remember.
It's important to keep your boxes, as it says something about the care you took with the units to a potential buyer for re-sale. This is especially important with things like turntables, which could be easily damaged if not shipped in their original packaging.
People like me are more likely to buy your used gear if you have the original box.