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).
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.