In a glass measuring cup, add the marinade ingredients and whisk to combine.
Remove ¼ Cup as a reserve for after the chicken is cooked.
Add the remaining marinade in a ziploc bag, add the chicken and seal shut.
Allow the chicken to marinate for 30 minutes in the fridge.
Turn the grill to high heat and allow to heat during the marinating process.
Thread bell pepper and onion on the skewers, followed by chicken and then topping off with more pepper and onion. *SEE NOTE.*
Turn off all but two burners and place the skewers on the grill as well as the lemon halves.
Turn the heat to medium.
Grill the meat, turning often, until grill marks appear.
Move to the warm but not hot side of the grill to finish cooking, about 15 minutes start to finish or until meat is cooked through.
Remove the meat to a platter and tent (not wrap) with foil for 5 minutes to allow juices to redistribute.
Drizzle with the reserved marinade and a squeeze of the grilled lemons and serve with tzatziki and pitas.
Making the Tzatziki Sauce:
Slice the ends off of the cucumber, slice in half lengthwise.
Using a spoon, slide the spoon down the inside of the cucumber to remove the seeds.
Finely chop the cucumber and place in two paper towels.
Wrap the towels and squeeze out the moisture as much as possible.
In a bowl, whisk together the cucumber and remaining ingredients into a tzatziki sauce.
Store in fridge until serving.
Notes
We heat all burners but turn off all but one to cook. I learned this with regular chicken breasts on America's Test Kitchen and have found it works well with kabobs to prevent dry meat.
*We've found that peppers and onions help the bottom and top piece of meat from drying out as well since they are protected by the veggies.