Soak the bamboo skewers in water for 30 to 40 minutes.

Place shredded cucumber in a fine mesh sieve and press out excess liquid; allow to drain for 15 minutes.

Combine yogurt, 2 cloves minced garlic, shredded cucumber, dill, salt, and pepper in a small bowl. Set aside in the refrigerator to chill.

Place chicken in a bowl and add the oregano, 2 cloves minced garlic, lemon juice, 2 tablespoons olive oil, salt, and pepper in a large bowl; stir thoroughly and set aside in the refrigerator to chill for 20 minutes.

Preheat an outdoor grill for medium-high heat and lightly oil the grate.

Combine tomatoes, diced cucumber, onion, 2 tablespoons olive oil, salt, and pepper in a bowl. Toss to combine.

Skewer the marinated chicken.

Grill the chicken skewers, squeezing with juice from 1/2 a lemon while cooking, until browned and no longer pink inside, about 5 minutes on each side.

Heat the pita bread on the grill until warmed through, 1 to 2 minutes on each side. Top each heated pita bread round with chicken from 1 skewer, 1 to 2 spoonfuls of yogurt sauce, and tomato salad. Top with feta cheese slices.