It seems that work on porting CyanogenMod 7 (Gingerbread) to the Amazon Kindle Fire is progressing well. The latest update from the XDA Developers forum is that a developer has got CM7 running on the Kindle Fire. Although a lot of work is still required to make it usable, just getting CM7 to boot is great progress.
Here’s an image (not brilliant quality) of CM7 running on the Fire. I look forward to hearing about further progress.