Cory Schenkel said on Facebook that his grandfather died on Sunday morning in the city of Boise, Idaho.
Kennedy won an Oscar in 1968 for Best Supporting Actor in Cool Hand Luke.
As well as the Naked Gun comedies, he also made memorable appearances in Earthquake and Airport 1975.
Mr Schenkel said the veteran actor had been in poor health since the death of his wife, Joan, more than a year ago, and had been in a hospice for the past month.
Kennedy's daughter, Shannon Kennedy Sullivan, said on Facebook that she hoped people would "remember all the stories he told and the dumb jokes he loved to tell".
"He could always get a laugh and I will miss those but will cherish the little strips of paper he use to write them on when the family would go out to dinner."
Kennedy was born on 18 February 1925 in New York where his father worked as a musician and his mother was a dancer.
He served with the US infantry during World War Two, winning several decorations, and in the 1950s he worked for Armed Forces Radio and Television before moving to Hollywood.
The 6ft 4in actor quickly became well known for playing tough-guy characters.
Mark Wahlberg, the lead in Kennedy's final film, 2014's The Gambler, and Mia Farrow, who starred alongside him in the 1978 film Death On The Nile, were among those who paid tribute to him.
Talk show host Larry King, Family Guy creator Seth McFarlane and actor and comedian Albert Brooks also shared their thoughts.

The two victims, both 15, are being treated in hospital. One remains in a critical condition.
The attack happened at Kingsdale Foundation School at about 15:30 GMT on Thursday.
The teenager accused of attacking the pair has also been charged with grievous bodily harm and possession of an offensive weapon.
He will appear at Camberwell Magistrates' Court on Saturday.

It was jockey Ryan Moore's second win in the race, having ridden O'Brien's previous winner Gleneagles, in 2015.
Barney Roy (7-2) was second ahead of third-favourite Al Wukair (11-2) in the first Classic of the 2017 Flat season.
Frankel's offspring Dream Castle and Colts Eminent were fifth and sixth respectively over the mile distance.
The 1,000 Guineas, for fillies, takes place at Newmarket on Sunday at 15:35 BST.
Churchill, last season's champion two-year-old, was sitting third at the halfway stage and managed to get the first run on his main rivals as he won by a length from Barney Roy.
Moore said Churchill, who was making his three-year-old debut at Newmarket, had "everything you want in a racehorse".
"He has a magnificent mind, he travels, has speed and loads of class," Moore told ITV Racing. "He was always racing comfortably - he was there a little bit early.
"He always feels like there's more when you ask him."
O'Brien added: "I was very worried about it being his first run. He's a big horse and we knew he would come on for the run. Ryan gave him a lovely ride.
"Everyone at home was very happy with the horse which is why we took the chance to come first time, so I'm delighted."
BBC racing correspondent Cornelius Lysaght
Am I the only one to feel a slight sense of anti-climax?
Obviously, Aidan O'Brien's latest record-breaking feat is a testament to the prolific skills and brilliance of himself, the rest of the Coolmore team, and, of course, of champion stallion Galileo, Churchill's father.
But as a race, I'm not sure. It was hardly the potentially epic showdown billed; the time wasn't great, and the first seven finished within about three lengths.
Churchill was good rather than sparkling, though it was his first run of the season. A word for Barney Roy who stumbled at a crucial moment late on.
US actor George Kennedy, who starred in movies including Cool Hand Luke and the Naked Gun series, has died at the age of 91, his grandson has announced.
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A 15-year-old boy has been charged with attempted murder after two pupils were stabbed at a school in Dulwich.
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Churchill, the 6-4 favourite, won the 2,000 Guineas at Newmarket to give trainer Aidan O'Brien a record-breaking eighth win in the Classic.