In 3-quart saucepan, heat gizzard, heart, neck, and 4 cups water to boiling over high heat. (Do not add liver; broth may taste bitter.) Reduce heat to low; cover and simmer 1 hour. Strain, reserving meat and broth. Pull meat from neck; discard bones. Coarsely chop neck meat and giblets. Cover and refrigerate meat and broth separately.
To make gravy, remove rack from roasting pan. Pour pan drippings through sieve into 4-cup liquid measuring cup or medium bowl. Add 1 cup giblet broth to roasting pan, and stir until brown bits are loosened; pour through sieve into drippings in measuring cup. Let stand 1 minute, until fat separates from drippings into 2-quart saucepan; skim and discard any remaining fat from drippings. Add remaining giblet broth and enough water to drippings in cup to equal 3 cups.
Into fat in saucepan, stir 1/4 cup all-purpose flour and 1/2 teaspoon salt; over medium heat, stirring, until flour turns golden brown. Gradually stir in drippings mixture and cook, stirring, until gravy boils and thickens slightly. Stir in reserved giblets and neck meat; heat through. Pour gravy into gravy boat. Makes about 3 1/2 cups.