Darn it. I had written a nicely detailed explanation of proper grip, te no uchi, and swinging mechanics, but aikiweb deemed to log me out, so it disappeared. Here's the short version.
1) Make sure your grip is correct. There are a myriad of details here and they're all important, so check with your sensei if you don't know.
2) Make sure you apply good te no uchi (the combination wringing-squeezing action Jun mentioned).
3) Don't try to apply power all the way until you want to stop. Let the sword coast for a small bit.
The one piece of advice that people have given with which I strongly disagree is just letting the sword fall. While this probably makes it easy to "cut" straight and stop the sword cleanly, your cut will be slow and ineffective, as well as close to your own body. Even those sword schools that do cut close (as opposed to the extended cutting style of kendo, the various Itto Ryu, the Hayashizaki iaido and battojutsu schools, among others) do not just let the sword drop. You have to actually swing down (out is a better way of thinking of it for a lot of people) applying force. Obviously over muscling it (using the wrong muscles or the right muscles in the wrong combination) is no good, but you still need to use the muscles of your lower body, back, and forearms.