Set oven control to broil. In medium bowl, mix turkey, water chestnuts, onions, teriyaki sauce and pepper. Shape mixture into 4 patties, each about 1/2 inch thick. Place on rack in broiler pan.
Broil patties with tops about 6 inches from heat 4 minutes. Turn patties; cook 6 to 8 minutes longer or until meat thermometer inserted in center of patties reads 165°F.
Place patties in buns with cucumber slices and lettuce leaves.