Put the olive oil, lime juice, soy sauce, allspice, sugar, thyme, salt, black pepper, cinnamon, nutmeg, garlic, scallions, jalapenos and ginger in a food processor and blend until smooth.
Put the chicken into resealable bags and evenly distribute the marinade among the bags. Seal tightly, thoroughly coat the chicken in the marinade and marinate, refrigerated, for at least 30 minutes but preferably overnight.
Prepare a grill for medium-high heat.
Grill the chicken skin-side down on direct heat, turning once halfway through, until both sides are browned, 10 to 15 minutes. Transfer the chicken pieces to indirect heat and grill until a meat thermometer registers 165 degrees F and the juices run clear, 15 to 20 minutes more.
Serve the jerk chicken with a salad of green leaf lettuce, halved cherry tomatoes, thinly sliced red onions, shredded carrots and sliced celery.