Combine first 8 ingredients in a large bowl. Divide mixture into 8 equal portions, shaping into 1/4-inch-thick patties. Spoon 1 tablespoon feta cheese onto each of 4 patties (do not spread to edges); top with remaining patties, pressing edges to seal well.
Coat grill rack with cooking spray; place on grill over medium-hot coals (350° to 400°). Place patties on grill rack, and grill, covered, 6 to 8 minutes on each side or until done.
Wrap pita halves in heavy-duty foil. Add foil-wrapped pita halves to grill rack, and grill 2 minutes or until pitas are warm, turning foil pouch occasionally.
Line each pita half with lettuce. Place 1 slice tomato and 1 patty in each pita half. Drizzle 2 tablespoons Yogurt-Cucumber Sauce over each pita half.