Mix Italian dressing, soy sauce, Worcestershire sauce, garlic, mustard, ginger, and brown sugar in a large container. Place chicken in the container and marinate in the refrigerator, 8 hours or overnight.

Preheat the grill for high heat.

Lightly oil grill grate. Discard marinade and place chicken on the grill. Cook until chicken juices run clear, about 30 minutes. Smaller parts will be done more quickly. An instant-read thermometer inserted near the bone should read at least 165 degrees F (74 degrees C).