Chrome: If you want to delete messages you receive on Facebook, you have to select the message, click the Actions button, and then select delete. It's a cumbersome process, but Facebook Fast Delete Messages is a Chrome extension that simplifies it down to one click directly from your inbox.
Facebook Fast Delete Messages adds a small red X next to your Facebook inbox so you can delete messages with one click (the default blue X is only for archiving messages). When you delete the message, the entire thread disappears forever. If you don't like hanging onto old messages, Facebook Fast Delete will come in handy.