To prepare souvlaki, combine the first 5 ingredients in a zip-top plastic bag; seal and shake to combine. Add chicken to bag; seal and shake to coat. Marinate chicken in refrigerator for 30 minutes, turning once.
Remove chicken from bag; discard marinade. Thread the chicken and zucchini, alternately onto each of 4 (8-inch) skewers.
Heat a grill pan coated with cooking spray over medium-high heat. Add skewers; cook 8 minutes or until chicken is done, turning once.
To prepare tzatziki sauce, combine cucumber, yogurt, 1 tablespoon lemon juice, 1/4 teaspoon salt, and 1 garlic clove, stirring well. Serve the tzatziki sauce with souvlaki.