There are two ways to delete private messages in your inbox.

1)
a) Go to an individual message and open it.
b) Look in the lower right below the message for a box that says "Delete this Message". Click it and the message will be deleted

2)
a) Go to your inbox
b) Check the check box to the left of the message topic. Check all the messages you want deleted.
c) Scroll to the bottom of that page to the pull down menu that will have a number (the number of messages you checked) and "Selected Messages". Pull down the menu and click the "Delete" radio button and then the "Proceed" button at the bottom of the menu.
d) Your messages will be deleted.