Working with one chicken breast at a time, place chicken inside a zip-type bag and pound the thicker part of the chicken with a meat mallet to create a more even piece. Transfer chicken breasts to a separate gallon size zip-type plastic bag.
Combine the remaining ingredients in a food processor and blend until mostly incorporated. Pour yogurt mixture over chicken, move the chicken around in the bag until well coated. Seal bag and marinate chicken in the refrigerator for at least 3 hours or overnight.
Build a medium fire in a charcoal grill, or heat a gas grill to medium-high. Brush grill grates with oil. Scrape excess marinade off chicken; season with salt. Grill chicken, turning once, until browned and cooked through, 3-4 minutes per side.