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New mother board has nothing to do with Lightroom.
If they keep the same hard disk, you are fine.
I have the Lightroom 4 upgrade and I know it only has a one pc install license and that Lightroom registers with the motherboard to determine how many pcs it has been installed on.
100% incorrect. Again, it is your hard disk, not your motherboard, where the LR registration is stored.
Lightroom doesn't use authentication. If you have your serial numbers, that's all you should need, and if they only replace the motherboard, then LR will still have the numbers and you shouldn't need to do anything (unlike with PS).