Combine all of the ingredients for the marinade in a bowl and whisk well. Taste to see if you have enough salt. (Make sure to taste before you add the chicken!)

Cut the chicken into about 1-inch cubes and add into a large zip-top bag (or a glass food storage container for marinating). Add the marinade and mix well to make sure all the chicken is coated well.

Let the chicken marinate for at least 2 hours or overnight. (Don't exceed 12 hours of marinating time.)

For the best results, cook the chicken kebabs on the grill.

If you will be using wooden skewers, make sure to soak them in water first for about 45 minutes.

Prepare the grill and preheat it to medium-high heat. (As always, it's best to have two temperature zones on the grill so if you need to move the chicken over to the cooler side, you can.)

Skewer the chicken into the soaked skewers, fitting 4-5 pieces onto each skewer.

Grill the chicken kebabs for about 4-5 minutes per side, turning them a quarter of the way. (Total time will be about 16-20 minutes, depending on the thickness of the chicken pieces.)

While chicken is cooking, lightly brush pita bread with olive oil and place them onto the grill to heat through. Slice the vegetables and place them onto a serving dish.

To serve chicken souvlaki, top each warm pita bread with some grilled chicken, sliced onions, tomatoes, and tzatziki sauce.