Combine giblets, except liver, neck, onion, celery, and salt in a small saucepan. (Set liver aside.) Cover with water; bring to a boil. Reduce heat; cover, and simmer 1 to 2 hours or until giblets are fork tender. Add liver, and simmer an additional 10 minutes. Drain, reserving giblet broth; discard onion and celery. Remove meat from neck; coarsely chop neck meat and giblets. Set aside.
Skim fat from pan drippings of roast turkey or chicken, being careful not to lose any of the brown sediment. Discard fat. Reserve pan drippings in roaster; add reserved giblet broth, stirring until sediment is loosened from bottom of roaster. Measure broth mixture and add water to equal 1 1/2 cups, if necessary. Pour mixture into a medium-size saucepan.
Add flour to giblet broth mixture, stirring until smooth. Cook over medium heat, stirring constantly, until thickened and bubbly. Stir in reserved neck meat and giblets. Add salt and pepper to taste. Serve warm with roasted chicken or turkey and dressing, if desired.