1In ungreased 8-inch square glass baking dish, mix all marinade ingredients; remove 1 tablespoon marinade and reserve. Add tuna to marinade; turn to coat. Cover and refrigerate at least 15 minutes but no longer than 2 hours.
2Spray grill rack with cooking spray. Heat coals or gas grill for direct heat. Remove tuna from marinade; discard marinade. Cover and grill tuna over medium heat about 10 minutes, turning once, until fish flakes easily with fork.
3Mix reserved 1 tablespoon marinade and the mayonnaise; spread on cut sides of buns. Layer lettuce, tuna, tomato and cucumber in each bun.