Combine first 4 ingredients in a large saucepan. Bring to a boil; cover, reduce heat, and simmer 2 hours, removing liver after 20 minutes to prevent overcooking. Remove from heat. Pour broth through a wire-mesh strainer into a bowl. Remove neck meat from bone; chop, and set aside. Chop remaining giblets, and set aside.
Stir 3 1/2 cups broth into reserved turkey drippings in broiler pan that turkey roasted in, or in a large saucepan. Bring to a boil. Combine flour and 1/2 cup water, stirring until smooth; gradually whisk into boiling broth. Add salt and pepper. Reduce heat to medium, and cook, whisking constantly, 5 minutes or until thickened. Stir in chopped neck meat and giblets; cook until thoroughly heated.