In a medium mixing bowl, combine the mustard, cayenne, and vinegar. Whisk together as you add oil in a thin stream.
Transfer the marinade to a large zip-top bag. Add the chicken pieces and seal the bag. Massage the bagged chicken well to distribute the marinade around each piece.
Place the bag on a tray in the fridge for several hours.
When ready to cook, heat an outdoor grill or stovetop grill pan. Remove the chicken from the marinade and place on the hot grates. Cook, turning once, until an internal temperature of 155 degrees is reached. Cover and let rest for ten minutes. Serve hot, cold, or at room temperature.
Note: the chicken in marinade may be frozen instead of step 3. Thaw completely before proceeding with step 4. Note regarding nutritional information: since you won't be consuming all the marinade, the caloric value will be significantly smaller. However, there's no scientific way to calculate this.