I've just spotted this question. It's a well-known open problem in discrete geometry. Unfortunately, I can't remember what it is called.
There is a paper that proves that no union of finitely many lattices can work. I'll try to remember what it is. I remember that it used a result of Mahler.
Added later: I think it was Mahler's selection theorem. Maybe that's enough of a clue for someone to reconstruct the proof (which I seem to remember wasn't hugely hard).