For the marinade: Combine olive oil, lemon juice, garlic, oregano, salt and pepper; place in a large re-sealable plastic bag. Add chicken pieces to the bag, re-seal, and massage the marinade into the chicken. Let marinate in refrigerator for at least 3 hours, and up to 5 hours, turning and massaging bag occasionally.
Prepare the sauce: Combine all ingredients and refrigerate for at least 1 to 2 hours to allow the flavors to develop.
Soak 6 wooden skewers in water for at least 30 minutes OR have 6 metal skewers ready.
Heat grill to medium-high and lightly oil the grate.
Remove chicken pieces from marinade and thread onto the skewers, discarding the unused marinade.
Grill kabobs for about 6 minutes per side, until chicken is no longer pink in the center.
Serve with tzatziki sauce for dipping, or make pita sandwiches.