Brush chicken with oil; sprinkle with 1/2 teaspoon salt and pepper. Place the chicken on grill rack coated with cooking spray; grill 5 minutes on each side or until done. Place chicken on platter; let cool 5 minutes. Slice each chicken breast half in half lengthwise, and slice chicken pieces crosswise into thin slices.
Combine chicken, cucumber, and next 6 ingredients. Sprinkle with remaining 1/4 teaspoon salt; toss to combine. Garnish with mint sprigs, if desired.