Blackberry systems can access POP mail servers for their users, then push it to their handheld device. They get in the middle. That's why you used to get POP mail pushed out on a BB, I suspect.
iPhones connect directly to the POP server. Of course, at times when BB network is down, this is a plus. Some POP/IMAP servers do have support for ActiveSync, which is the MS technology that pushes Exchange email to devices like the iPhone. SmarterMail, for example, now supports ActiveSync. It may also be an option on GMail - not sure about that.
Anyway, you may want to check again with your mail provider and see if they offer ActiveSync or some similar solution too. If they do, set up using the "Exchange" option on the iPhone and you might get the push feature you're looking for.