Option 1: Use the Forward Busy/No Answer feature. It will send callers straight to the greeting. This also allows for the receptionist and Auto Attendant to do blind transfers to extensions. Calls will then follow the Forwarding and "roll into voicemail". Set FWD Busy/NA on all phones to the voicemail (Ex: On KXTD systems press intercom + FWD/DND + 5 + 165).
Option 2: Change the Incomplete Handling Status for each mailbox. Go into every mailbox and press 4 and then 4 again. You will be presented with several different options. Disable "Operator Transfer Status" and "Returning to Top Menu Automated Attendant Status". Now, since "Leaving a Message Status" is the only one enabled, it will skip callers past the menu and get them to the mailbox greeting.
Of course, none of this applies to the useage of the Voicemail Transfer (VTR) button, which will always send callers directly to a mailbox greeting.