In a nonreactive bowl, combine the yogurt with all the remaining ingredients except the chicken. Stir to blend well, add the chicken breasts, marinate, covered and refrigerated, at least 3 hours, or overnight for richer flavor.
For the sauce, heat the olive oil in a medium saucepan over moderate heat. Saute the garlic, onion, and cucumbers until softened, about 5 minutes. Stir in the curry, cumin, and cayenne. Add the tomatoes and simmer until juicy, about 10 minutes.
Preheat a broiler grill, grill pan, or outdoor grill.
Remove the chicken from the marinade and set aside (if making kebabs, skewer the chunks). Add 1/2 cup of the marinade to the tomato sauce. Bring to a boil, lower the heat, and simmer 5 minutes.
Puree the sauce in a blender or food processor. Return to the saucepan and season with 1/2 teaspoon salt. Keep the sauce warm.
Cook the chicken breasts over a hot grill until cooked through but still juicy, about 5 minutes a side (less for kebabs). Alternatively, cook the chicken breasts under a preheated broiler, 4 to 5 minutes a side.
Transfer the chicken to a platter and serve with the warm sauce.