Possible Clues:

• Silk gets brief, thanks to French cheese
• A lustrous silk
• River swallowed up fine material
• Substantial returns on cheese cloth
• Rejected heavy Greek product for lightweight stuff
• 'Fine stuff gets fortissimo, in a teat (7)'
• Material that's thick wrapped around cheese
• Stiff shiny material
• Fabric put over large cheese
• Old president keeps iron and a more delicate material