Wright, who took on the role at the end of 2010, had declined an offer to extend his contract, citing a desire to pursue other interests.
"I appreciate the offer to extend my contract but after much consideration I feel it's the right time to move on and look at other opportunities," the 57-year-old said in a statement issued by New Zealand Cricket (NZC).
NZC chief David White said: "John has been an outstanding servant for cricket in New Zealand over a long time and it is disappointing to lose someone of his calibre.
"We were keen to see him continue his head coach role, however understand and respect his decision to look for another challenge.
"We are keen to fill this critical role as soon as possible but are conscious that we need to take the time to find the right candidate who can take us through to the 2015 ICC [ Images ] Cricket World Cup."
A former New Zealand captain, Wright amassed over 9,000 runs in Tests and One-Day Internationals as a left-handed opening batsman before taking up coaching with stints with English county side Kent and India [ Images ] from 2000-2005.
As New Zealand coach, Wright oversaw the team's run to the one-day World Cup semi-final last year and December's Test win over Australia [ Images ] in Hobart, the first on Australian soil since 1985.
New Zealand kick off their West Indies tour with two Twenty20 [ Images ] matches in Florida [ Images ], the first on June 30, followed by five One-day matches and three Tests in the Caribbean in July and August.