Heat gas or charcoal grill. In 1-quart saucepan, mix teriyaki glaze, apple jelly and gingerroot. Heat to boiling, stirring constantly, until jelly is melted. Spoon half of the mixture into small bowl or custard cup; stir in cilantro. Set aside to serve with chicken.
Cover and grill chicken over medium heat 15 to 20 minutes, turning and brushing with apple-ginger mixture during last 10 minutes of grilling, until juice of chicken is clear when center of thickest part is cut (170°F). Serve with reserved apple-ginger mixture.