The three sides will play two matches apiece in the series, which will help New Zealand and Bangladesh prepare for the ICC Champions Trophy in England.
"Gone are the days, we hope, of being starved of the matches we deserve," said Cricket Ireland's Warren Deutrom.
Dates and venues for the series - in May 2017 - are yet to be confirmed.
"Rather than getting nine or 10 ODIs in the four years between World Cups, we aim to play at least that number annually," added chief executive Deutrom.
"For years now, we have been asking for more matches to test our players, to raise the profile of our sport, to bring in new fans, and to attract more investment."
Ireland have a busy schedule of home fixtures arranged for this summer, with two games against Sri Lanka in June, a five-match series against Afghanistan in Belfast in July plus two encounters with Pakistan in August.
The Irish will also take on South Africa and Australia in South Africa in late September.
Earlier this week, the ICC announced it had given both Ireland and Afghanistan more than £340,000 to help the two countries arrange more matches as they attempt to improve their world ranking and secure places at the 2019 World Cup.
Next year's New Zealand games will see Ireland coach John Bracewell face his native country.
"We are now getting the volume of cricket that the team needs to take their cricket to the next level," he said.
"With home advantage in at least 13 ODIs we'll certainly have plenty of opportunities to test ourselves against quality opposition."
Ireland defeated Bangladesh at the 2007 World Cup but then lost against them four years later in India.
The Irish lost their first one-day international against New Zealand at the 2007 World Cup in Guyana and were beaten again in their only other meeting in Aberdeen a year later.
Ireland will host New Zealand and Bangladesh in a one-day tri-series next year as they continue their efforts to play more games against full members.