First investigate how the cabinets are attached to the wall and where those fasteners are located.
You should be able to loosen the fasteners in the cabinets near the bottom of the cabinets enough to allow you to pry the plywood from behind the cabinets. First be sure this can be done safely. Remove the microwave. Turn off the power to the outlets and loosen them from their boxes. This way they can be twisted so they can be pushed through the holes they now occupy. You will
damage the drywall but that's easy enough to repair.
Pry the board from the bottom, you should be able to cause the nails to give up that may be behind the cabinets.
Any nails behind the cabinets will be only where the studs are, at worst about 16 inches apart. There is probably only one nail per stud hidden behind the cabinets. That's not enough nails to cause any major issues.