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.
Tap Change Voicemail Password. Enter the new password (4-6 digits) then tap Done. If requested, enter the current password. Re-enter the new password, then tap Done. If that doesn’t work, contact your phone carrier to change your passcode or PIN. Or if your phone carrier has an account app (like Verizon’s My Verizon App), you can update

The greeting section does not work. Call your own number and listen to the automated voice tell you whether you have messages or not. Then wait for further instructions. It will tell you to press four for personal options and then you press three to change or listen to your previous greeting.
It's in the upper right corner of the screen. Now, if you are prompted to type in your password when you are checking voicemail, you should enter the password you just set. When checking voicemail from your iPhone, usually you don't have to type in the password again because your iPhone has automatically sent the password to your carrier already.