Add ground beef, sausage and liver to a large saucepan over medium heat. Season with salt and pepper and cook, crumbling with a wooden spoon, until browned.
Add flour and stir. Add onion, bell pepper, celery and sauté for 2 minutes. Add garlic, Cajun seasoning, cayenne, oregano, thyme and bay leaves and stir to combine.
Add rice and chicken broth. Bring to a boil, cover, reduce heat to low and simmer for 20 minutes. Remove from heat all allow to rest for 5 minutes. Remove lid, gently fluff rice with a fork, reapply lid and rest for 5 more minutes.
Garnish with green onion and parsley before serving.
Chicken Livers: If you don't like chicken livers you can leave them out, or substitute small uncooked, chopped pieces of chicken thighs, or stir in chopped cooked chicken at the end. For the Instant Pot: Turn to Saute setting and brown meat. Add and saute veggies and spices. Add rice and 2 cups of chicken broth. Cook on manual/high pressure, for 4 minutes with a 10-minute natural release. Remove lid, gently fluff rice with a fork, reapply lid and rest for 5 more minutes. Garnish with green onion and parsley before serving.