1. Preheat grill to medium-high (350° to 400°). Combine first 8 ingredients and 1 teaspoon kosher salt in a large bowl, and reserve 1/4 cup mixture in a second bowl. Add chicken to first bowl; marinate 15 minutes.
2. Meanwhile, toss cucumber, radishes, and dill in reserved dressing. Season with kosher salt and black pepper to taste. Garnish, if desired.
3. Remove chicken from marinade, discarding marinade. Grill chicken 5 minutes on each side or until done. Serve with cucumber-radish salad.