For raita, stir together 3/4 cup yogurt, cucumber, lemon juice and 1/4 teaspoon salt in a small bowl; set aside.
In another small bowl, stir together remaining 1/2 cup yogurt, remaining 1/2 teaspoon salt, ginger, garam masala and cayenne. Place chicken in a baking dish and spread yogurt mixture over top. Cover with plastic wrap and refrigerate 30 minutes.
Heat gas grill to medium-high or prepare charcoal grill with medium-hot coals. Lightly coat grill with nonstick cooking spray or oil. Grill chicken, covered, about 20 minutes over direct heat, turning every 5 minutes. Then move chicken to indirect heat for 5 minutes or until internal temperature measures 160° on an instant-read thermometer. Serve with raita.