When you go to the message listing on Facebook, it says ‘+ New Message’ in the button at the top and the messages each have an ‘x’ to the far right. But that ‘x’ will only archive the message, it doesn’t delete it.
To actually delete messages click on any message and the top buttons will be ‘Messages’ and ‘Actions’ then click on ‘Actions’ and you should see the option to delete messages listed there.
This is not at all like email that many of us have become used to. It almost seems to be an intentional effort to hide the feature, maybe the developers just wanted to be different.
Hope this helps and please post a comment if you are still having any issues deleting messages on Facebook.