In small bowl, combine preserves, lime juice, soy sauce and garlic; mix well. Spoon 1/4 cup sauce into 1-cup measuring cup. Stir in cornstarch until smooth; set aside. Refrigerate remaining sauce in bowl until serving time.
Heat oil in medium skillet over medium-high heat until hot. Add chicken; cook 5 to 7 minutes or until browned, stirring frequently. Stir in cucumber, carrot, peanuts, gingerroot and garlic. Stir in reserved sauce with cornstarch. Cook 2 to 3 minutes or until bubbly and thickened, stirring constantly. Spoon mixture into shallow dish. Cover; refrigerate 30 minutes or until cooled.
Spoon 2 tablespoons chicken mixture onto center of each lettuce leaf. Fold sides of lettuce in toward center; roll up like burrito. Secure with toothpicks. Place seam side down on serving platter. Serve immediately, or cover and refrigerate until serving time. Serve wraps with sauce.