This add-on for Firefox will add a ‘Facebook – Delete All Messages’ option to Tools dropdown menu that deletes all listed conversations!
Here is a little more about the add-on from the developers:
If your current tab is at “www.facebook.com/messages” location and you click the delete all messages option from Tools, a confirmation dialogue will appear and ask you to delete everything. Afterwards, you will see your messages disappear! If you are at the wrong location and click the delete all option, a new tab will open and you will be redirected to the correct URL.
Want to learn more? You can pick up Facebook – Delete All Messages from the Firefox Add-ons website.