Twistory is a free tool to back up your tweets. . It goes beyond a simple data dump, though, by plotting your tweets in an easy-to-import iCal feed.
Twistory is really simple to use, and doesn't require authentication. Just feed it your Twitter username—or any Twitter name, really—and it kicks back an iCal feed link that you can push into iCal, Google Calendar, or any calendar app that accepts the .ics format. Start using it with Google Calendar before you reach Twitter's 3,200 tweet backup limit, and you'll have a search-ready, exportable database of your tweets.
Twistory is a free service, doesn't require a sign-up to use.