Twitter's direct message (DM) system allows you to easily communicate privately with your followers. You can delete DMs individually, but Twitter's website doesn't give you a simple option to mass-delete all of them at once. You'll have to turn to an outside app and give it permission to access your account. Be aware, however, that when you delete a DM you've sent to someone, it's also removed from that person's inbox.
Visit the InboxCleaner website and click "Sign in with Twitter."
Click "Allow" to grant InboxCleaner access to your account.
Click "Delete all messages," "Delete messages from user" or "Delete messages by content" to wipe out your messages. As of April 2011, InboxCleaner is charging for its services to delete all of your DMs at once; if you don't want to pay, you can only delete up to 20 messages per day, so it may take you a few days to clean out your inbox.