The appellants said the US advised that any money they paid to counsel or experts in America would be a forfeit, and anyone who took such money would be guilty of a crime.
But the appellants said they needed help from US lawyers and experts to defend the extradition application.
Batato, Ortmann and Van Der Kolk applied to the Court of Appeal on August 7 and Dotcom did so a fortnight later.
A half-day hearing was set down for last Tuesday but the hearing took longer than expected and the Court of Appeal reserved its judgment for a few days.
"We heard more than three hours of high speed argument," the Court of Appeal said in its judgment today.
"By the end of the hearing, it was apparent to the court that giving a judgment that did justice to the arguments presented was an impossible task in the available time, given the volume of material presented."
The appellants said before they were eligible for extradition, there had to be a prima facie case they had committed an extraditable offence.
A prima facie case relies on facts presumed to be true until disproved.
The appeal discussed issues around the US-New Zealand extradition treaty and the Vienna Convention, which concerns diplomatic relations between nations.
The four men said they needed to be able to call evidence from an American expert as to their relevant intentions when running Megaupload.
"For example the appellants are charged with conspiracy to breach copyright. There is an issue whether that is comprised in the offence of conspiracy to defraud, contained in the Extradition Treaty," the Court of Appeal judges added when describing the appellants' argument.
"Natural justice requires that the appellants be able to call evidence from an American expert to challenge the prosecutor's evidence, and to mount an argument that the United States has breached the duty of candour required of a requesting state."
The Megaupload group were accused of running the site in a way that made them party to copyright breaches.
The appellants argued it was necessary to call technical evidence from American experts as to the way similar cloud storage websites were run, and how Megaupload ran and responded to copyright holders' complaints.
The US argued the appellants' upcoming hearing on the North Shore would give them a chance to mount their natural justice argument anyway.
The appellants could address any remaining natural justice concerns by seeking judicial review after next week's district court hearing, the US claimed.
The US also said the extradition process required it to show only a prima facie case that the appellants had committed a treaty offence.
And in this case, the US argued, the relevant crimes alleged were offences under New Zealand law or the extradition treaty, not US law.
The Court of Appeal judges said the appellants' submissions regarding technical evidence were "vague" and failed to explain how evidence of cloud storage industry practice affected their eligibility for extradition.
The court also said it did not have the resources "urgently to resolve a dispute which was not straightforward and involves fundamental disagreement between the parties".
For three reasons, the Court of Appeal decided to reject the appeal.
First, the judges said the district court had been managing the issue appropriately and in line with the extradition treaty.
Second, the court said it had to be careful about "interfering" in district court pre-hearings.
"Micro-management of pre-trial procedure by higher courts risks delaying and fragmenting the criminal process," the Court of Appeal said.
Third, the Court of Appeal said it was satisfied going ahead with the stay application and extradition hearing together on September 21 would not deprive the appellants of adequate remedies.
The Court of Appeal said if a district court judge decided the US had deprived the appellants of American expertise they needed, then "inevitably" that judge would have to adjourn the extradition hearing.
The appeals were dismissed and the appellants were ordered to pay the US's costs.
Dotcom commented on the appeal on Twitter, where he had 465,000 followers.
"Today we'll find out if I get a fair extradition hearing or if the New Zealand judiciary will transform itself into a US owned dancing bear," he wrote as the judgment loomed.
"As expected...Dancing bear," he said later.
