Trim all visible fat and tendons from chicken breasts, then cut each bread into about 6 crosswise strips. (I made the strips about 1 inch thick.) Put chicken strips into large Ziploc bag or plastic container with a tight-fitting lid.
Combine olive oil, lemon juice, vinegar, garlic, oregano, and thyme and stir together with a whisk or spoon. Pour marinade into Ziploc bag or plastic container, seal, and squeeze bag or turn container over and shake gently so all pieces of meat are well-coated with marinade. Put into refrigerator and marinate 8 hours or more.
To get ready to grill, remove chicken from refrigerator and let it come to room temperature, then drain chicken pieces and thread onto skewers. I put six pieces on each skewer, and folded over each piece of chicken so it wouldn't spin around on the skewers. Spray grill with non-stick spray or mist with olive oil, then preheat gas or charcoal barbecue grill to medium high. (You can only hold your hand there for a few seconds at that heat.)
Grill skewers about 12-15 minutes, turning often. Souvlaki is done when the chicken is slightly browned and firm, but not hard to the touch.