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1. Set a yogurt strainer, or regular strainer lined with a double layer of dampened cheesecloth, over a bowl. Add the yogurt to the strainer and let drain, in the refrigerator, for 4 to 6 hours.
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2. Remove the drained yogurt from the refrigerator (it will be quite thick) and discard the liquid. Transfer the yogurt to a large nonreactive bowl. Add the garlic, ginger, onion, lemon juice, cayenne, salt, black pepper, and cumin seeds and whisk to blend. Set aside while you prepare the chicken.
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3. Remove and discard the skin from the chicken pieces, then rinse the pieces under cold running water. Drain and blot dry with paper towels. Make 2 or 3 deep slashes, to the bone, in each piece. Add the chicken to the marinade in the bowl and turn the pieces to coat. Cover and let marinate, in the refrigerator, for 4 to 6 hours.
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4. Preheat the grill to high. (Note: For best results, use charcoal.).
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5. When ready to grill, oil the grill grate. Remove the chicken from the marinade and arrange on the hot grate. Grill, turning with tongs, until nicely browned on the outside and the juices run clear when the tip of a skewer or sharp knife is inserted in the thickest part of a drumstick or thigh, 8 to 10 minutes per side (16 to 20 minutes in all).
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6. Transfer the chicken to serving plates or a platter and serve immediately, garnished with onion, tomato, cucumber, and radish slices and lemon wedges.
