Twedemption is a paid service, at one time of $5.99 USD, to go in and remove all the direct messages from your inbox.
Now I have seen other apps from the company, so I was not as nervous about providing my username and password to Twitter as others might be. But, for a first time user, there is no help on the main page, no contact information (just a blank form link) and nothing about how it works.
Also, you must have a lot of direct messages stored up in your inbox to need this service. While the web interface to Twitter takes forever to click delete for each one, this is a full sweep move and might get rid of something you want to keep. Just remember that.