Hey carnut,
1. Click on the Private messages, you'll see the total of messages that you have on the righthand side. click the box beside it.
2. At the bottom of the page, you'll see a Selected Messages box, select the item Marked as Read and click the Go button, it should make all the messages in that folder (in this case, the Inbox) to be marked as read.
Cheers,
madcatlancelot