Boiling water and dishwasher detergent. Stir it up and let sit for a little while. You will be amazed at the gunk. I had to do this to mine, and I think I used about a tablespoon of detergent and filled the carafe with boiling water.

I'll do that! I noticed it was better after a trip thru the dishwasher, so another treatment might help more. Thanx.

Dump a bit of baking soda in the carafe, and add a dish cloth, fill 3/4 full with hot water(not boiling). Put on the lid and shake. I clean my coffee carafe, my coffee pot, my tea pot, and any thermoses this way.

Dump out the water. The cloth will be right there so remove it and rinse well

Denture cleaning tables also work. The ones made for coffee stains.

I second kris' remedy. My stainless steel drip coffee maker carafe has been cured with the boiling water and dishwasher detergent remedy. After this treatment it comes out like brand new again.

baking soda and peroxide also works.