Jones was executed at the Arizona State Prison Complex in Florence, about 60 miles south of Phoenix, The Arizona Republic reported.
Jones was convicted of killing two men in a robbery at the Moon Smoke Shop and three women and a man at the Firefighters Union Hall social club. The killings took place two weeks apart in June 1996.
He also admitted killing a retired Republic executive, Richard Roels, in August 1996 at Roels' home in Phoenix. Jones received a life sentence for that crime.
The killing of Roels led to Jones' arrest after police tracked the use of the dead man's credit cards. Jones was linked to the Tucson killings by a man named David Nordstrom, who said he had been the getaway driver in the smoke shop robbery, and the killers were his brother, Scott, and Jones.
Scott Nordstrom remains on death row. David Nordstrom served a short sentence for armed robbery.
Jones' lawyer, Dale Baich, an assistant federal public defender, said he declined a special last meal.
"It's just another meal and there's nothing special about the day to me," Baich quoted his client as saying Tuesday night.