Okay, I am still learning how to use the new software. Sorry for the slow response.
To delete private messages, go to "MY RC" by hitting that bottom off the top of the page.
Then hit "List Messages" in the left hand column, under the blue "Private Messages" heading
Check the boxes for the messages you want to delete. The boxes are the right-most column on the message listing.
Go to the bottom of the message listing, just above "Display Options" and click the "Selecting Messages" menu area and select "Delete".
Somehow, I think there should be an easier way, but I haven't found one.