The former England all-rounder had been in the role for a little under a year, initially on an interim basis as Shane Bond's replacement, before taking it on full-time.
The constant travel away from his young family had been the main reason behind his decision to give up the post, the former Otago coach said.
"At first I thought I could balance the responsibilities but I no longer think that's feasible. I need to be closer to my family," the 38-year-old said.
"I had an amazing time with the team. It's probably been the best time I've had in cricket, either as a player or as a coach."
Australia skipper Steve Smith's side depart for their short tour of New Zealand on Saturday, for an itinerary that begins with the first of three one-day internationals next Wednesday.
The sides play two tests, the first in Wellington from February 12-16 and the second in Christchurch from February 20.
New Zealand's next international commitment after that series will be the World Twenty20 tournament in India in March.