1. For marinade: Mix first 7 ingredients in large shallow baking dish. Add chicken and turn to coat. Cover and refrigerate at least 4 hours or overnight, turning chicken occasionally.
2. Prepare barbecue (medium heat). Preheat oven to 350°F. Wrap tortillas tightly in foil and place in oven. Heat 3 tablespoons oil in heavy large skillet over medium-high heat. Add peppers, onion and garlic and sauté until tender, about 4 minutes. Season to taste with salt and pepper. Keep warm in oven.
3. Remove chicken from marinade and grill until golden brown and almost cooked through, about 3 minutes per side. Layer 1/4 cup cheese, 1 chili, then another 1/4 cup cheese onto each breast half. Grill until chicken is cooked through and cheese melts, about 3 minutes.
4. Arrange 1 chicken breast on each tortilla. Top each with 1/4 of sautéed pepper mixture. Wrap tightly. Place tortilla, seam side down, in center of each plate. Garnish burritos with cilantro and serve, passing salsa separately.