Here's an easy way to make cleaning a fireplace screen part of your regular cleaning routine. Once or twice a season, use a cleaning solution of 1/8 cup liquid dishwashing detergent per quart of water to remove the caked on dirt.  Gently scrub the fireplace screen with a soft bristle brush and follow up by wiping with a lint free cloth to avoid rusting.  Polish the brass section of the fireplace screen with a brass cleaner and a lint free cloth.