Long story short: Apple has its own voicemail service, when the iPhone is activated on an existing AT&T cell phone number, the AT&T voicemail is killed dead for the cell phone and the Apple voicemail takes over. I subscribed to unified voicemail, where my cell and home phones share a mailbox, and when a message is received on either line, both are notified and I get the message emailed to me as an audio file. Activating the iPhone removed the cell phone part of that system.
Logging into my messaging system account on the web, it showed the cell phone number was still linked. However, when I called the cell phone from work, I got a message saying I could not leave a message because the mailbox had not been set up yet. So I called my cell phone number from my cell phone, which put me into voicemail management, and recorded greetings. But all this did was give me a separate mailbox for the cell phone, which is not what I wanted.
So, online this morning I removed the cell phone from the unified voicemail account, waited 10 minutes, then re-added it. And recorded new greetings. Voila! it works now.
There is no reason I can think of why AT&T's stellar support staff could not have figured this out. This, on top of not getting a signal at work, is prompting me to look elsewhere for phone service.