Heat grill to medium-high. Form the beef into four 1/2-inch-thick patties and season with 1/2 teaspoon salt and 1/4 teaspoon pepper.
Grill the burgers to the desired doneness, 4 to 5 minutes per side for medium, basting with the teriyaki sauce during the last 2 minutes of cooking. Grill the English muffins until toasted, 1 to 2 minutes per side.
Sandwich the burgers in the muffins. Serve with the tomatoes and cucumbers.