Grate the cucumber and squeeze out excess liquid with a clean towel until very dry.
Whisk yogurt, lemon juice, olive oil, garlic, dill, mint, salt, and pepper in a large bowl until smooth.
Fold in the chicken, dried cucumber, red onion, and feta until evenly coated.
Cover and chill for 15 minutes to let the flavors meld.
Serve stuffed into warm pitas, over crisp greens, or with crackers and veggies.