Two of the men, aged 45 and 49, appeared in the Hamilton District Court today, jointly charged with burglary.
They were both granted interim name suppression, one was granted bail while the other was remanded in custody until he can find an appropriate residence.
The man on bail entered a not guilty plea and would now reappear in court in March.
The 49-year-old accused was remanded in custody without plea until February 2, however his lawyer indicated an application for bail would likely be made prior to that date.
The third accused is set to appear after today's lunch adjournment.
Meanwhile, the Goldie painting has not been recovered.
"Police seek the public's help with any information that may lead to the recovery of this painting," a police media release said.
"If you have any information, please contact Police on 105 and quote file 210103/2961.
"Alternatively, you can contact Crime Stoppers on 0800 555 111."
Goldie's most expensive piece, A Noble Relic of a Noble Race, of Ngāti Manawa chief Wharekauri Tahuna, sold for $1,337,687 at an International Art Centre auction in Auckland in 2016.
Director Richard Thomson said he'd sold another version of the stolen painting in 2008 for a then-record of $454,000.
"So that is a million-dollar-plus artwork there in today's market. I've sold dozens of Goldies, and that is a really good example of his work, has all the hallmarks.
"I am quite disturbed by [the burglary]. These are hugely significant national treasures. Owners are guardians but the nation owns them, really."
The stolen painting was likely done between 1933 and 1938, when Goldie was in his sixties.
While Goldie's earlier works tended to fetch the highest prices, Thomson said the 2016 record was painted in 1941.
Despite the high valuation, Thomson said he thought it would be "worthless" in the hands of thieves.
"There is absolutely no market for it now in the wrong hands. It is a foolish thing to do and all they will get is bad karma.
"My advice is return as quick and safe as possible."
