On the iPhone, go into voicemail and tap "Greeting". Record a few seconds of something and tap stop. Before you tap save, use iFuntastic or SFTP to navigate to /var/root/Library/Voicemail on your iPhone. You'll see a file called "Greeting. amr". This is the audio you just recorded. Delete that file and replace it with your custom message which should then be renamed to "Greeting.amr". Tap "Play" on the iPhone to verify your message works. If it plays, tap "Save" and your custom message will be uploaded to AT&T's servers.
Offer other contact information such as a cell phone or email address if you will check it (i.e. on vacation). DON’T: Be too informal, even if you are creating a humorous message. Give a prolonged sales pitch in your voicemail greeting message. Waste a caller’s time with an excessively long greeting before they can leave their message.

No you can’t. I just called and asked for it to be turned off completely and was told it is impossible. It’s aggravating to get conflicting information. Obviously either the CS Team member who wrote the previous post, or the employee I just spoke to on the phone, doesn’t know what they’re talking about.
You might also consider giving them the option to skip your voicemail message completely by pressing another key (again, check with IT). You also want to make callers aware what you will return their call, instead of telling them that you will try to return their call--and let them know when you are likely to do it, if possible. If necessary, provide additional or emergency information in your voicemail greetings. My dentist, for example, lists his cell phone number for emergencies.