Place the chicken pieces in 1 or 2 large re-sealable plastic bags and add the rest of the ingredients. Seal the bag and shake to mix. Refrigerate at least 1 hour or overnight.
Heat grill to medium heat, or until the coals are lightly covered with ash. Oil the grill.
Cook the chicken, skin side up, for 20 minutes. Turn the pieces and cook an additional 15 to 20 minutes, or until a meat thermometer inserted reaches 155 degrees and the chicken is fork-tender and the juices run clear when a thigh is pricked with a fork. (It should measure 180 degrees F on an instant read thermometer). Alternately, roast the chicken in a preheated 400 degree F oven for 35 to 40 minutes.
Serve 1 bird hot and reserve the other for Days 2 and 3.
Brush the vegetables with olive oil and sprinkle with salt and pepper.
Place everything on the grill over medium heat and cook, turning frequently, for 10 to 15 minutes, or until fork tender. Serve hot, reserving 1/3 of the vegetables for Day 2 and two ears of corn for Day 3.