Ok, so here are the symptoms. Your iPhone is setup to get/send email and appointments with your corporate MS Exchange server. You use MS Outlook as your email and calendar application on your Windows machine at the office.
If you get a meeting invitation it shows up on your iPhone calendar notifications as well as Outlook. If you accept it on your iPhone everything is fine and Outlook understands it's been accepted. However if you accept it from Outlook your iPhone won't realize it's been accepted and will still show it as a notification and an unaccepted meeting in your calendar. You then have to accept it from your iPhone which sends an additional accept notification to the meeting creator causing them to think you are an idiot.
As I searched the web for a solution I found convoluted answers about Exchange not being setup correctly, MS and Apple bugs, etc.
There was another clue to my issue. Two other co-workers with iPhones were not having this behavior so it was somehow specific to my setup.
It turns out this is a bug in the iPhone if you have Outlook setup to keep meeting requests in your inbox after accepting/declining. You see by default (and most users keep their default settings) Outlook will remove a meeting invite from your inbox once you respond. At that point the only way to find that meeting is in your calendar and no longer in your inbox. This can be annoying if a meeting request had an attachment and you didn't remember the date of the meeting and need to open the attachment. So whenever I get a new work PC with Outlook one of the first things I do is change that default setting to leave meeting requests in the inbox.
Apparently the iPhone uses that information to determine if you've accepted/responded/denied a meeting request. This appears to be a bug in the iPhone as it assumes the user hasn't changed their email settings in Outlook.
To get the iPhone to recognize that a meeting request has been responded to in Outlook so it stops asking you to accept/decline, do the following in Outlook (This should work for Outlook 2003 and 2007, not sure about other versions): Go to Tools->Options->Email Options->Advanced Email Options. Look for the item called "Delete meeting request form Inbox when responding" and make sure it is selected.
If it was already selected then you have a different issue then I did. This has solved my problem 100%.