Start a charcoal fire with the coals in the center of the grill.
Rub the chicken pieces with salt, pepper and cayenne to taste.
Mix mayo, cider vinegar, and sugar.
Rub the chicken with the mayo mixture, reserving a small amount to baste the chicken with at the end of its cooking.
Let the chicken sit for about 30-45 minutes while charcoal gets ready to cook.
Grill the chicken skin side down for 30 minutes, to the outside of the grill, not directly over the hot coals. After 30 minutes, turn the chicken and continue to cook until done, turning occasionally as needed. Time will vary depending on the size of your pieces and how hot the fire is.
When the chicken is done cooking, baste the chicken with the reserved marinade.
You can substitute a package of any bone in chicken pieces you like for the whole chicken!