Prepare outdoor grill with medium-hot coals, or heat gas grill to medium, or heat oven broiler.
Meanwhile, place cut-up peach in food processor. Pulse with on-and-off motion until peach is chunky. Combine peach, preserves, vinegar, sugar, soy and ginger in medium-size microwave-safe glass bowl. Microwave, uncovered, on high power for 2 minutes or until peach is slightly softened, stirring once after 1 minute.
Mix together cornstarch and water in small bowl until well blended. Stir into peach sauce. Microwave, uncovered, on high power for 2 minutes or until slightly thickened, stirring once after 1 minute. Reserve half the sauce to serve with chicken.
Remove skin from chicken legs. Salt and pepper both sides of legs. Arrange legs in single layer in 13 x 9 x 2-inch microwave-safe baking dish, with thickest part of each chicken leg facing the outside of the dish. Loosely cover the baking dish with plastic wrap. Microwave on high power for 5 minutes. Turn chicken legs over. Loosely cover again with plastic wrap. Microwave for 5 minutes. Let chicken stand, covered, for about 5 minutes. Brush remaining half of sauce over legs.
Grill legs 6 inches from heat or broil 2 inches from heat for about 4 minutes on each side, until chicken becomes dark in color, but not burnt, and instant-read thermometer inserted in thickest part without touching bone registers 170°F. Serve with the reserved half of the sauce on the side.