1. In a large resealable plastic bag, combine the first eight ingredients; add chicken. Seal bag and turn to coat; refrigerate for at least 1 hour.
2. Drain and discard marinade. Using long-handled tongs, moisten a paper towel with cooking oil and lightly coat the grill rack. Grill chicken, covered, over medium heat or broil 4 in. from the heat for 4-6 minutes on each side or until a meat thermometer reads 170°. Spread mayonnaise over bottom of rolls; top with chicken, lettuce and cheese. Replace roll tops. Yield: 6 servings.