Combine 1/4 cup olive oil, lemon juice, 2 cloves minced garlic, oregano, and 1/2 teaspoon salt in a large resealable bag. Add chicken, coat with the marinade, squeeze out excess air, and seal the bag. Marinate in the refrigerator for 2 hours.
Mix yogurt, cucumber, 1 tablespoon olive oil, vinegar, 1 clove minced garlic, and 1 pinch salt together in a bowl. Refrigerate tzatziki sauce for flavors to blend, 1 to 2 hours.
Preheat an outdoor grill for medium-high heat and lightly oil the grate. Soak wooden skewers in a bowl of water for about 15 minutes.
Remove chicken from marinade and thread onto the soaked skewers. Discard unused marinade.
Cook the skewers on the preheated grill, turning frequently until nicely browned on all sides and chicken is no longer pink in the center, about 8 minutes per side.