I read quite a bit about this when considering a Kindle purchase. I read about people complaining of fully charged Kindles going through the X-ray machine and becoming de-charged. I also read about some screens scrambling after going through the X-ray machine. I, personally, have never had either of these things happen, nor has anyone I know. I also have never had a TSA agent ask me to take my Kindle out of my bag.
One thing that had happened, that surprised me: on domestic flights, I have had flight attendants consider a Kindle an “electronic device” and ask me to turn it off before taking off. Not on international flights though.
So the bottom line is: it’s probably a good idea to turn your Kindle completely off before going through security to protect it from the X-ray machine. However, you don’t need to take it out of your bag. Have it easily accessible in case a super-zealous screener asks to see it, but don’t worry about actually having it out. And don’t get mad if the flight attendant tells you to turn it off before takeoff.