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.
Unfortunately, I have been asked a few times to remove my Kindle from my bag for screening. It seems to be rather hit-or-miss. I’ve not had any problems with the machinery de-charging it though.