I had a question like this a while back. See if any of the answers help you:
http://stackoverflow.com/questions/114953/how-to-get-email-and-their-attachments-from-php
postfix + maildrop was the solution I ended up taking, It routes the emails through to a PHP script when it arrives and in my case, the PHP does something with the attachment. But I only needed to read each email once. If you need to be able to browse all the emails, you either need to store the results of maildrop or find another solution.
If you need to full access to all emails, POP and IMAP are popular choices because they do work. I'm not sure why you're against them.