In glass container with fitted lid, whisk together olive oil, lemon juice, mustard, garlic, oregano, thyme, salt, and pepper. Add chicken breast, coating on both sides with the marinade. Seal container and refrigerate for 1 to 2 hours.
When ready to cook, remove chicken from refrigerator and bring to room temperature while you preheat oven to 350 F. Bake chicken for 45 minutes, or until instant-read thermometer registers 165 F . Allow chicken to cool completely and refrigerate to chill for minimum 2 hours.
In small bowl, dress feta with olive oil and oregano. Cut chicken into 12 bite-sized cubes. Assemble skewers by placing pieces of cucumber, tomato, a cube of chicken, red onion, pepper, feta, and olive on each skewer. Pack in airtight container and place in your cooler.
Combine yogurt, grated cucumber, lemon juice, and mint. Add to small Mason jar with lid and pack in cooler or picnic basket. Bring a spoon so picknickers can drizzle the dressing over skewers.