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You need Google Apps for Domains
http://www.google.com/a/
Basically gmail accounts for your domain. It's free and comes with 2gb of storage for each account.
So you can have
president@organization.com as a gmail account with no forwarding required.
You only need the standard (free) edition.
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I briefly looked at this. I don't see how this is much different than what I mentioned. The account is already set up with godaddy, so no matter what forwarding needs to be set up there, correct? I don't want to require that people log into gmail, as great as I think it is.
Here's an additional thought. If people aren't logging into a position specific gmail account I don't really have a way to catch communications that they send out. Only incoming emails will be caught through either forwarding to a catch all, or forwarding to a gmail address, which forwards again to a personal email.
Edit: I'm looking at Google Apps now, looks like there's more going on here than I thought.