Heat the olive oil in a large pot over medium heat. Add the chicken pieces and cook until lightly browned on all sides.
Add the onion, carrots, and celery. Cook for 4 to 5 minutes until the vegetables begin to soften. Stir in the garlic and cook for 1 minute more.
Pour in the chicken broth and bring the mixture to a simmer. Let it cook for 10 minutes.
Add the orzo and continue simmering for 8 to 10 minutes until the pasta is tender.
Remove the pot from the heat. Stir in the lemon zest, lemon juice, spinach, and dill. Season with salt and black pepper to taste. Let the spinach wilt for a minute before serving.