If you need to cool them below ambient as well as heat them the peltiers are probably the answer. I keep trying to talk you out of cooling them because it will add a lot of cost and complexity.
I think I would start out with an aluminum plate the size of the board as a heatsink. Then peltiers and the small metal squares. I would use some type of insulator for the white squares - maybe pcb material.
I think I would use two PC power supplies for the peltiers so you would have a + and - 12 volt supply. If we are on the right track we can talk about temperature control.