Prepare the chicken: Combine all ingredients in a large freezer bag. Rub all around, coating chicken as evenly as possible. Let marinate for 20 minutes or up to 24 hours, whatever time you have. Make tzatziki: Place the yogurt in a medium bowl. Quarter your cucumber lengthwise and cut out the seeded area. Coarsely grate your cucumber onto a clean dishcloth and wring it out as well as you can. All of that liquid makes for a soggy sauce. Add wrung-out cucumber to yogurt bowl along with lemon, vinegar, dill, garlic, salt and pepper and stir to combine. Taste and adjust seasonings. Chill until needed.
Prepare salad components: Arrange lettuce, onion, tomatoes and cucumbers on a platter. If desired, toss onion with a squeeze of lemon and/or dash or two of vinegar and salt so that it mellows/lightly pickles while it rests.
Cook chicken: Heat your grill to high heat or a grill pan to medium-high. Remove chicken from marinade and grill on first side until dark lines appear, about 5 minutes, then flip and grill for 3 minutes more on second side. Cooking times will vastly range by the thickness and coldness of your chicken, as well as variations between grills. Transfer cooked chicken to a cutting board, let rest for a couple minutes while you quickly heat/toast your pitas on the grill.
Cut chicken into thin strips. Cut pitas into wedges. Add to platter with vegetables and let everyone dig in. Repeat all summer.