After a few cracked boards I paused to curse and kick at the pallet, and then sat down and decided to puzzle out a different way of doing it without having to pull out each nail. And here is how it is done:
Step 1: Cut slats off the support board
Step 3: Trim the nails
Step 4: Cleaned boards
You need all the support pieces from both pallets. Six in this case. No pallet ever seems to be made the same, so if your pieces are not the same length, trim them down to match. In my case I had a pallet about half a foot longer than the other one.