The Kindle Fire ships with Amazon's own custom version of Android v2.3 Gingerbread, and as you can probably guess they have tweaked it to suit themselves. They've locked out their competitors' reading apps, rendered a number of other apps incompatible, and so on. But none of that matters any more, because now you can install a completely Amazon-free firmware.
A hacker going by the name of JackpotClavin has just posted a set of instructions as well as all the files you'll need in order to install it. But it's not nearly ready enough for the end user. CM7 is running okay, but not everything works yet. Audio is still wonky and at least one user is having stability issues.
You might also want to wait until the hack is reversible, too. Right now there's no going back, but JackpotClavin is planning to release a restore image which you can install and (hopefully) get back to the original factory settings.
This version of Cyanogen Mod is based on Gingerbread but if you want something more modern, wait for a bit. JackpotClavin said that he has shifted his attention to Android v4.0 Ice Cream Sandwich. And that is definitely worth waiting for.
So should you do it? I don't have my KF anymore, so I cannot. But this might be worth it. One the one hand, you'll lose some Amazon services like Instant Video. But on the other hand you'll also gain Google apps, Android Market, and generally turn the KF into a standard Android tablet. That might be worth it, given the specs.
BTW, there's one detail that hasn't been settled yet but is definitely pinning down: Bluetooth. The Kindle Fire has a Bluetooth chip inside which cannot be used under the stock Amazon firmware. I'm hoping that enabling Bluetooth is a software issue, not hardware. This would add a lot of value to the KF.