Heat the olive oil in a large pot over medium-high heat. Add the andouille sausage and cook until browned, about 3 minutes. Remove and set aside.
In the same pot, add the chicken and season with salt and pepper. Cook until browned, about 5 minutes. Remove and set aside with the sausage.
Add the onion, bell pepper, and celery to the pot. Cook for about 5 minutes, stirring occasionally, until softened.
Stir in the garlic, smoked paprika, thyme, oregano, and cayenne. Cook for 1 minute until fragrant.
Pour in the diced tomatoes and chicken broth, then stir in the rice. Bring to a boil, then reduce heat, cover, and simmer for 20 minutes.
Return the sausage and chicken to the pot, then add the shrimp. Stir everything together and cook for another 5 minutes until the shrimp is pink and cooked through.
Garnish with sliced green onions and serve hot.