To prepare chicken, combine first 10 ingredients in a bowl. Place 1/4 cup tahini mixture in a small bowl; reserve. Place remaining tahini mixture and chicken in a large ziplock plastic bag; seal, turning to coat chicken. Refrigerate 8 hours or overnight.
Preheat grill to medium (350°F to 450°F). Coat grill grate with cooking spray. Remove chicken from marinade; discard marinade. Sprinkle chicken with salt. Arrange chicken on grill grate; cover and cook 4 minutes on each side or until done. Arrange chicken on a platter; drizzle with reserved 1/4 cup tahini mixture.
To prepare the salad, combine cucumber and next 6 ingredients (through 1/4 teaspoon salt) in a bowl; toss gently to combine. Serve cucumber salad with chicken and rice.