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.

Erhan Havaleoglu, 36, killed Leighanne Cameron, 29, by strangling and stabbing her in Mid Calder on 28 October 2015.
He must serve a minimum of 20 years before he can apply for parole.
Sentencing him at the High Court in Glasgow, Judge Johanna Johnston said he was guilty of a "cruel, merciless and extremely violent murder".
In August, a jury at the High Court in Paisley found him guilty of murdering Ms Cameron.
The court heard the attack happened after Havaleoglu discovered she was in a relationship with another man.
The judge told Havaleoglu: "Leighanne Cameron was much loved and had everything to live for."
She said Ms Cameron's injuries had been "catastrophic".
Havaleoglu attacked the mother-of-two shortly after she returned from work.
The killer had been spending the day looking after their children before he struck.
He then fled to Edinburgh and it is thought that on his way there he got rid of the murder weapon and the clothes he had been wearing.
Ms Cameron's body was discovered by her new boyfriend Philip Kilkenny.
Mr Kilkenny called police and revealed Havaleoglu had told Leighanne "she'd never get to love another man".
The court heard the "reality" of Ms Cameron's new relationship had come to "the full attention" of Havaleoglu shortly before the murder.
Ian Duguid, defending, said it came from a conversation the killer had with one of his young children.
The QC said this discovery was suggested to be "the catalyst" for the fatal attack.
Mr Duguid: "This is an event that pushed him beyond what his previous behaviour had been."
Havaleoglu had also been accused of assaulting three other women, but he was cleared of those charges at the trial.
Det Ch Insp Raymond Brown, of Police Scotland, said: "Leighanne's death has been absolutely devastating for her family and those close to her.
"Her five-year-old daughter and two-year-old son have been robbed of their loving mum in the most horrific of circumstances.
"Havaleoglu has callously refused to admit responsibility for Leighanne's murder ever since, despite extensive evidence against him."
He added: "Unfortunately, nothing can undo his actions that day but hopefully the outcome at court provides some form of comfort and closure to Leighanne's family and friends."
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.
[NEXT_CONCEPT]
A man has been given a life sentence for murdering his estranged wife at her West Lothian home while their two children slept upstairs.