1. Combine the cucumber, radish, and red onion in a medium bowl. Whisk together the next 4 ingredients (through salt) in small bowl until sugar is dissolved; pour over vegetables, and toss gently. Cover and refrigerate until ready to serve.
2. Prepare grill. Lightly spray the grill rack with cooking spray.
3. Sprinkle salt and pepper on chicken, and pat into flesh. Grill chicken, covered, over medium-high heat for 4 minutes on each side or until internal temperature is at least 165°. Serve chicken with a side of the cucumber salad.