Heat gas or charcoal grill. In medium bowl, mix tomatoes, onion, vinegar, salt and 2 teaspoons of the oregano. Let stand at room temperature until burgers are ready to eat.
Meanwhile, in small bowl, mix remaining 1 teaspoon oregano, the cheese, lemon peel and oil; set aside.
In large bowl, sprinkle beef with garlic-pepper blend; gently mix until well blended. Shape mixture into 6 balls. Using end of thumb, make a deep depression in each ball, but do not push all the way through. Spoon about 1 tablespoon cheese mixture into each depression. Gently shape beef around cheese mixture, sealing it inside. Press balls to form patties, 1/4 inch thick and about 5 inches in diameter.
Place patties on grill over medium heat. Cover grill; cook 12 to 16 minutes, turning once, until meat thermometer inserted in center of beef reads 160°F.
Place burgers on bottoms of buns; spoon tomato mixture over burgers. Top with tops of buns.