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.
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.