Not all of the presidents died, not all were murdered, not all were even president in the years in question. It isn't even that every twenty years, a president dies in office--because other presidents died in office (or came near death) in between these deaths.

Only one, though. 7 of the 8 US Presidents who have died in office to date were elected in zero years:

1840 - William Henry Harrison, died of pneumonia in 1841
1860 - Abraham Lincoln, assassinated in 1865
1880 - James Garfield, assassinated in 1881
1900 - William McKinley, assassinated in 1901
1920 - Warren G. Harding, died of heart attack in 1923.
1940 - Franklin D. Roosevelt, died of cerebral hemorrhage in 1945
1960 - John F. Kennedy, assassinated in 1963.

The only other President to die in office was Zachary Taylor. Elected in 1848, he died in of cholera in 1850.

From what I've read, the "curse" did not specify cause of death.