Just in case you prefer Mozilla's Firefox browser, there is hope in sight: Mozilla said that it is preparing Firefox 8 for approval in Amazon's Appstore, which would indicate that the software could be available for the Kindle Fire soon. Of course, Amazon has the final say whether the browser will be offered for download or not.
There is not much information beyond the note that Mozilla's developers are working through "any approval issues" for Firefox Mobile 8, which indicates that the browser could become available shortly through Amazon. Not all Amazon Android Appstore apps have been approved for the Kindle Fire: there are 600 apps available in total, but only 362 can be downloaded to the Kindle Fire.
Meanwhile, Firefox 9 is currently on stage for a Beta 4 build and is scheduled for a December 20 release.