Soak 8 (10-inch) wooden skewers in water 30 minutes. Heat gas or charcoal grill. Place 1/2 cup peanut sauce in medium bowl; set aside. Cut chicken crosswise into 1/2-inch thick slices. Thoroughly coat chicken slices with sauce in bowl; sprinkle with 1/4 teaspoon salt. Cover and refrigerate 30 minutes.
In another medium bowl, stir together cucumber salad ingredients; set aside.
Remove chicken from bowl; discard peanut sauce. On each skewer, thread 4 to 5 chicken slices. Place skewers on grill over medium heat. Cover grill; cook 6 to 8 minutes, turning once, until chicken is no longer pink in center.
To serve, divide cucumber salad among 4 serving plates. Top with chicken skewers and toppings.