(CNN) - It may be months before Mitt Romney announces who will join him on the GOP presidential ticket, but speculation surrounding the likely nominee's choice of running mate is already well beyond fever pitch.
Joining the discussion Monday: former Vice President Dick Cheney, the last Republican to act as the country's second-in-command. Making some of his first public remarks following a successful heart transplant in late March, Cheney drew on his own experience selecting running mates in advising Romney.
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"I've been involved in a couple of vice presidential searches, some more successful than others," Cheney said at an event for students at the Washington Center. His remarks were carried by C-SPAN.
In 2000, Cheney led the vice presidential search for then-candidate George W. Bush that culminated in his own selection as running mate.
He continued, "The thing that I think is important to remember is the decision you make as a presidential candidate on who your running mate's going to be is the first presidential level decision the public's going to see you make. It's the first time you're making a decision you're going to have to live with. It gives the public a chance to see you operate and see what you think is important, what kind of individual you choose to serve as your running mate, what are the criteria."
Cheney said making a vice presidential decision in order to leverage key voting blocs was a mistake, saying it ignored the position's most vital duty.
"I think the single most important criteria has to be the capacity to be president," Cheney said. "That's why you pick them. I think lots of times in the past that has not been the foremost criteria."
Romney would be smart to ignore pundits handicapping the decision, Cheney said.
"As you watch the talking heads out there now, they're saying you've got to pick a woman, a Hispanic, someone from a big state. They're all interesting things to speculate about but it's pretty rare that the election ever turns on those kinds of issues."
In his remarks, Cheney also offered details about the night he received his heart transplant. The former vice president had been on the waiting list for a donor heart for 20 months.
"On a Friday night about midnight I got a phone call," Cheney said. "I knew I was getting to the top of the list."
He continued, "We got in the car and drove to the hospital. Checked in there. About seven o'clock in the morning they began the operation. It took about five or six hours."
Cheney said he remained in the hospital for nine days following the procedure.
"I'm feeling very well, and very fortunate," Cheney said of his current state of health. He added he felt "great gratitude to the individual, family that donated the heart I was privileged to receive."