Place the chicken breasts in the slow cooker. Add the diced onion, sliced carrots, sliced celery, minced garlic, chicken broth, oregano, salt, and black pepper.
Cover and cook on low for 6 hours or until the chicken is fully cooked and tender.
Remove the chicken from the slow cooker and shred it with two forks. Return the shredded chicken to the pot.
Stir in the orzo, lemon juice, and lemon zest. Cover and cook on high for 20 to 25 minutes until the orzo is tender.
Stir in the fresh spinach until it wilts. Ladle the soup into bowls and top each serving with chopped parsley and crumbled feta.