This question is probably to Blender gurus (maybe it was asked before but I did not find a reference).
Is there settings (in Blender\3DXchange, or some other application) to have UV maps generated the way, so that each side of the object is skinned individually.

Here's, as an example, a difference between iClone cube primitive and the one exported in Blender

If you want a quick UV with all the faces flat, use the "Smart UV Project" It lays out the UV a little differently, but it does the trick. You can then re-arrange the UV islands if you want to.

Thank you rampa and sw00000p. I'll give it a try.