NEW YORK — Two American hikers being held in an Iranian prison got a big surprise one day after their exercise routine: Instead of being blindfolded and led back to their cell, they suddenly heard the words, “Let’s go home.”

That’s what a diplomatic envoy from Oman told them before whisking them away to the Tehran airport — and freedom, the two men said Sunday at a Manhattan news conference.

“After 781 days of prison, Shane and I are now free men,” a jubilant Joshua Fattal announced, hours after he and Shane Bauer landed at Kennedy International Airport.

Safe on U.S. soil, the two spoke for the first time in public about their ordeal of more than two years at the hands of Iranians — accused of spying for their country by illegally walking across the Iran-Iraq border.

They say they simply got lost while hiking with another American, Sarah Shourd, who was released last year.

The three paid a brutal price for their adventure, they said.

“Many times, too many times, we heard the screams of other prisoners being beaten and there was nothing we could do to help them,” Fattal said.

Added Bauer: “How can we forgive the Iranian government when it continues to imprison so many other innocent people and prisoners of conscience?”

Bauer was himself beaten and Fattal forced down a flight of stairs, Shourd told reporters.

And though their families wrote them daily letters, they had to go on repeated hunger strikes to receive the letters, the men said.

The two managed to hold on to reality by reading letters sprinkled with news of what was happening in the world, Bauer’s mother, Cindy Hickey, said.

Eventually, they were told — falsely — that their families had abandoned them.

Until their release, the last direct contact family members had with Bauer and Fattal was in May 2010, when their mothers were permitted a short visit in Tehran.

“Solitary confinement was the worst experience of all of our lives,” Fattal said. “We lived in a world of lies and false hope.”

But on Sunday, hope filled a media-packed conference room at Manhattan’s Parker Meridien hotel as the two 29-year-olds walked in, surrounded by relatives. A smiling Bauer put his arm around Shourd — now his fiancee.

He had proposed to her while they were both imprisoned, seeing each other only an hour at a time no more than once a day. He formed an impromptu engagement ring out of the threads from his shirt.

Fattal and Bauer were freed last week under a $1 million bail deal and arrived Wednesday in Oman, greeted by relatives and Shourd.

The men’s families told the AP on Sunday that they don’t know who paid the bail.

But the hikers do know who appeared at Tehran’s Evin prison to take them to freedom. That was the big surprise.

They had just finished their brief daily open-air exercise and expected, as on other days, to be blindfolded and led back to their 8- by-13-foot cell. Instead, the prison guards took them downstairs, fingerprinted them and gave them civilian clothes. They weren’t told where they were going.

The guards then led them to another part of the building, where they met a diplomatic envoy from Oman, who spoke the magic words, “Let’s go home.”

Within hours, the prison gates opened and the Americans were driven to the airport, then flown to Oman, a tiny Persian Gulf nation that had helped negotiate their release and is a U.S. ally.

The following days made for “the most incredible experience of our lives,” Fattal said.

Shourd was with the families to greet the two on the tarmac at a royal airfield in Oman’s capital, Muscat. At about 20 minutes to midnight Wednesday, Fattal and Bauer bounded down the plane steps — very thin and pale, but in good health.

In prison, they had kept in shape physically and mentally by lifting water bottles, discussing books and asking each other questions, family members said. And they ripped slivers of cloth from prison blindfolds to secure their sandals so they could run for exercise.

By Sunday, their returning energy was visible; they were feeling better and better each day, Hickey told the AP.

The first hint of a turnaround in the case came last week when Iranian President Mahmoud Ahmadinejad announced the two could be released within days. But wrangling within the country’s leadership delayed efforts. Finally, Iranian defense attorney Masoud Shafiei secured the necessary judicial approval Wednesday for the bail — $500,000 for each man.

Iran’s Foreign Ministry called their release a gesture of Islamic mercy.

A beaming Shourd told reporters Sunday: “Shane and Josh and I are beginning our lives again, and there are so many new joys that await us; I’ve never felt as free as I feel today.”

The couple haven’t yet made any wedding plans, she said.

Free and on home soil, Fattal and Bauer sharply rebuked Iran, declaring that they were detained because of their nationality, not because they might have crossed the border from Iraq.

“From the very start, the only reason we have been held hostage is because we are American,” Fattal said. “Iran has always tied our case to its political disputes with the U.S.”

They said they may never know if they actually stepped across a border that is not clearly marked amid wilderness.

The hikers’ detention, Bauer said, was “never about crossing the unmarked border between Iran and Iraq. We were held because of our nationality.”

The irony of it all, he said, “is that Sarah, Josh and I oppose U.S. policies towards Iran which perpetuate this hostility.”

But when they complained about their treatment, they said, the Iranian guards cited how U.S. authorities at the military base at Guantanamo Bay, Cuba, dealt with suspected terrorists there.

The men’s saga began in July 2009 with what they called a wrong turn into the wrong country. The three say they were hiking together in Iraq’s relatively peaceful Kurdish region along the Iran-Iraq border when Iranian guards detained them.

The two men were convicted last month of espionage and illegally walking into Iran, and were sentenced to eight years in prison. Shourd was charged but freed before any trial.

The three have always maintained their innocence.

During the news conference, the men took turns reading prepared statements. They didn’t take questions from reporters.

Fattal said he wanted to make clear that while he and Bauer “applaud Iranian authorities for finally making the right decision,” they “do not deserve undue credit for ending what they had no right and no justification to start in the first place.”

The two countries severed diplomatic ties three decades ago during the hostage crisis, when American diplomats were held for 444 days at the former U.S. Embassy in Tehran after it was stormed in 1979 by militants backing Iran’s Islamic Revolution. Since then, both have tried to limit the other’s influence in the Middle East, and the United States and other Western nations see Iran as the greatest nuclear threat in the region.

Since her release, Shourd has lived in Oakland, Calif. Bauer, a freelance journalist, grew up in Onamia, Minn., and Fattal, an environmental activist, is from Elkins Park, Pa., a Philadelphia suburb.

Bauer and Shourd were living in Damascus, Syria, when Fattal came to visit and the three went hiking.

On Sunday, the men’s families told reporters that they hadn’t made plans for what they would do next — except for carving out some private time together. They would not divulge their destinations in the coming days.