The Church called the decision "plain silly" and warned it could have a "chilling" effect on free speech.
It had hoped the 60-second film would be screened UK-wide before Christmas ahead of the new Star Wars film.
The agency that handles adverts for the cinemas said it could offend those of "differing faiths and no faith".
The advert features the Christian prayer being recited or sung by a variety of people.
They include refugees, a grieving son, weightlifters at a gym, a sheep farmer, a gospel choir and the Archbishop of Canterbury, the Most Reverend Justin Welby.
The advert was passed uncut by the British Board of Film Classification and given a "U" certificate, as well as receiving clearance from the Cinema Advertising Authority.
However, the Digital Cinema Media (DCM) agency, which handles British film advertising for the major cinema chains, Odeon, Cineworld and Vue, refused to show the advert because it believed it would risk upsetting or offending audiences.
In a statement, DCM said it had a policy of not accepting political or religious advertising content in its cinemas.
It said that "some advertisements - unintentionally or otherwise - could cause offence to those of differing political persuasions, as well as to those of differing faiths and indeed of no faith," and that "in this regard, DCM treats all political or religious beliefs equally".
"This advert is about as offensive as a carol service or church service on Christmas Day," he said.
"Let the public judge for themselves rather than be censored or dictated to."
The Reverend Arun Arora, director of communications for the Church of England, said: "We find that really astonishing, disappointing and rather bewildering.
"The prospect of many families attending the release of the new Star Wars film had seemed a good opportunity to launch the advert and a new website justpray.uk to promote prayer ahead of Christmas.
"The Lord's Prayer is prayed by billions of people across the globe every day, and in this country has been part of everyday life for centuries."
'Make people think'
He added: "In one way the decision of the cinemas is just plain silly, but the fact that they have insisted upon it, makes it rather chilling in terms of limiting free speech."
He encouraged people to visit the website, watch the film and make up their own minds "as to whether they are upset or offended by it".
Stephen Slack, the Church's chief legal adviser, warned the banning of the advert could "give rise to the possibility of legal proceedings" under the Equality Act, which bans commercial organisations from refusing services on religious grounds.
The refusal to show the advert is likely to reignite a debate about the place of religion and faith in the public arena, especially Christianity, and whether freedom of expression for believers is being stifled.
"I don't think people know a lot about Christianity these days anyway, and the opportunity to share the Lord's Prayer in a cinema environment would make people think - and realise that Christians come in all shapes and sizes."
But Terry Sanderson, president of the National Secular Society, said: "The Church of England is arrogant to imagine it has an automatic right to foist its opinions upon a captive audience who have paid good money for a completely different experience.
"The Church does not hesitate to ban things that it deems inappropriate from its own church halls - things like yoga. The cinema chains are simply exercising the same right."