Blend soy sauce, lime juice, garlic, cumin, paprika, oregano, 1/2 teaspoon pepper, and oil in a blender.
Put chicken in a large sealable bag and add marinade. Seal bag and marinate, chilled, 8 to 24 hours.
If using a charcoal grill, open vents on bottom and lid of grill. Light a large chimney starter full of charcoal (preferably hardwood). When coals are lit, dump them out along opposite sides of bottom rack, leaving a space free of coals (the size of the quartered chicken) in middle. When you can hold your hand 5 inches above the grill rack directly over coals for 3 to 4 seconds, coals will be medium-hot.
If using a gas grill, preheat all burners on high, then reduce heat to medium-high.
Discard marinade, then pat chicken dry. Oil grill rack, then grill chicken over area with no coals (or over a turned-off burner), skin side down first, covered, turning over once, until cooked through, 30 to 35 minutes (add charcoal to maintain heat).
note: If you aren't able to grill outdoors, chicken (quartered) can be roasted in middle of a 500°F oven in a 13- by 9-inch roasting pan with 1 cup water 30 minutes, then tented with foil and roasted until browned and cooked through, about 15 minutes more.