mailbox unavailable the server response was 5.7.1 client does not have permissions

Exchange 2007, Windows Server 2003 R2
I am using in 2005 in an application we have.  I am able to send mail when I am logged into the network as domain admin and logged into the application as any email user, but users are not able to send when logged into the network as themselves.  
In active directory I made the necessary users members of Exchange Organization Administrators and now they have permission to send mail.
I would rather make them members of a more secure group that still grants the permissions necessary.  Any suggestions?
Who is Participating?
I use System.Net.Mail and use the IP address of the exchange server.  Set the exchange server to accept relay from internal IP addresses.

To prevent even the possibility of spammers using this, I created a 2nd SMTP server on Exchange, and didn't make it publicly visible.  I use this to send all outgoing SMTP messages.
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