Here's how the procedure was described to me by a 3rd party tech, and it's a procedure I've down quite a few times.
Assuming the Audix is the Definity Audix 3.2 (rather than the external one) look for the little button on the card. Press it in and hold it there for a few seconds until the display starts flashing. I believe it displays "shutdown" during this time but I could be wrong.
If you try to log in to the Audix during this time it will let you know that it's shutting down. Try it out of hours sometime, it's not a big deal. Holding down the button for a short while again should reboot it, but you'll have to check the manual for that one. As all the voicemails are stored on a hard drive none of that will be lost.
Once the Audix is down, flick the power switches on the back of the cabinets and it's all turned off. Don't forget to save the translations first though (obviously), and do a "status period" to make sure there's no periodic maintenance running. It wouldn't matter if there is, I just don't like the idea of powering down a G3 whilst it's in the middle of a maintenance run.
Bringing it back up is just a case of making sure the flash cards are in their sockets and turning the power back on, on the back of the cabinets. Both Definity and Audix come back in just a few minutes, plus the status is always displayed on the screen so you know what's going on.
It is very nerve racking the first time you do it though. Maybe it might be worth the money to have a tech come down for an hour when the power gets turned back on ?
Pete