3/17/12 Update: Shortly after I posted this, Amazon released a software update for the Kindle that makes it very simple to remove items from your Kindle’s carousel. Just touch and hold the item you’d like to remove, and you’ll get a contextual menu that gives you three options:

The options are:
Click your desired option and you’re done. Anyway, props go out to Amazon for quickly responding to consumer demand on this problem. I’m preserving the old content in this post below.
Got an embarrassing book you don’t want people to see when you’re showing off your Kindle Fire? Amazon hasn’t made it easy to remove those books from your carousel, but it is possible. Here are the steps:

Deleting books from the carousel on your Kindle Fire isn't as easy as it sounds!
NOTE: Before proceeding, keep in mind you’ll have to set up everything again on your Kindle (this includes email, Facebook access, apps you may have downloaded and set up, etc.).
2) Go to Tools (by hitting the gear icon in the top-right of the Kindle) → More → Device → Reset to Factory Defaults. This will remove all of your personal data from your Kindle and power off the device.
3) Power up your Kindle Fire. This could take a few moments as the device resets itself to factory settings.
4) Re-register your Kindle again by following the prompts on the screen.
It’s frustrating not to have full control over what shows up in the carousel. Hopefully the cumbersome process outlined above will be updated by an upcoming Kindle Fire release from Amazon.
Be heard... Be the first commenter!