Whisk the yogurt, garlic, lemon juice, lemon zest, and salt together in a medium bowl.
Place the chicken in a 6-quart slow cooker. Pour over ⅓ cup Greek yogurt sauce over the chicken, tossing to coat. Cover the remaining yogurt marinade and refrigerate while the chicken cooks.
Cover the chicken and cook on high for 3-4 hours or on low for 5-6 until the chicken is tender and cooked through.
Remove the chicken from the cooking liquid. Slice into strips or shred with two forks. Drain all but ¼ cup cooking liquid. Add the chicken back to the slow cooker, then pour over ⅓ cup of the reserved Greek sauce marinade, tossing to coat. Gently warm though on high for 5 minutes.
Serve with the remaining ⅓ cup Greek sauce marinade on the side for dipping or drizzled on top.
