Put chicken breasts in a container and cover with teriyaki marinade. In a separate container, put mushrooms and sliced zucchini and remaining teriyaki marinade. Put both containers in the refrigerator for approximately 30 minutes.
Put zucchini and mushrooms on enough aluminum foil to be able to wrap tight. On separate aluminum foil put chicken breasts making sure that there is plenty of room for all breasts on the foil.
Grill chicken breasts until done and zucchini and mushrooms until soft (approx. 30 minutes). Pour the vegetable mixture on top of the chicken breasts, cover with mozzarella cheese and continue to grill until cheese is melted.