Whisk all marinade ingredients together in a large bowl.
Chop chicken into bite-size chunks and toss with marinade.
Pour marinated chicken into a large sealable plastic bag and squeeze out all of the air.
Refrigerate for 2-3 hours, or overnight (recommended) for the best texture and flavor.
While chicken is marinating, prepare the tzatziki sauce.
Slice the peeled cucumber in half, lengthwise.
Using a small spoon, remove all of the seeds from the center.
Grate half the cucumber and place into a medium-size bowl.
Finely dice the other half of the cucumber and add to the bowl.
Mix in all other ingredients, cover and refrigerate until chicken is ready.
Heat grill to medium-high heat.
While grill is heating-up, thread the chicken pieces onto skewers.
Cook chicken skewers for about 10-12 minutes for breasts (10-15 minutes for thighs), turning several times.
Test with a meat thermometer and pull the chicken off the grill as soon as it reads a safe 165 degrees F. internal temperature.
Serve with spicy tzatziki sauce and soft pita bread, if desired.