For roux, in a large heavy Dutch oven stir together flour and oil until smooth. Cook on medium-high heat for 5 minutes. Reduce heat to medium. Cook and stir for 10 to 15 minutes or until roux is reddish-brown in color (the deeper the color, the richer and more flavorful the gumbo). Stir in celery, sweet pepper, onion, and garlic; cook for 5 minutes more, stirring occasionally. Add sausages and cook until sausages are lightly browned.
Add chicken, the water, salt, cayenne pepper, and black pepper to sausage mixture. Bring to boiling; reduce heat. Cover and simmer about 1 hour or until chicken is tender and no longer pink (170 degrees F for breasts; 180 degrees F for thighs and drumsticks).
Skim off fat. Remove chicken; cool slightly. When chicken is cool enough to handle, remove meat from bones; discard bones. Coarsely chop chicken and return to sausage mixture in Dutch oven. Cook for 2 to 3 minutes or until chicken is heated through. If desired, serve with rice.