Randy Scheunemann, McCain's main foreign policy advisor, was the chief lobbyist for Georgia's government under Saakashvili until late last year, which is why any Cheney influence in the Georgian imbroglio is probably also McCain influence.
McCain has been much more anti-Russian than Bush, who, until recently always referred to Putin as a close friend, and who supposedly bonded recently with Medvedev. McCain is the one who proposed driving Russia out of the Group of 8, and keeping it out of the WTO. He's been consistently more confrontational than Bush, Rice or Gates.
I'd argue that McCain's campaign has a central interest in the Great Game, as well, the strategic contest to control Central Asi. Further, McCain thinks he can win the election if we begin to have major tensions with Russia. And, look who's advising him on foreign policy: Georgia's primary lobbyist in Washington, who's earned over a million dollars in contracts with Saakashvili's government.
It's a natural for McCain; it's wag the dog time. If Americans can be scared by a confrontation with Moscow, then the former POW, darling of the defense industry, will have a major advantage over Obama. Obama has no military background, and reason--as in let's talk this over; let's calm down--comes across to many Americans as weakness. Bill Clinton famously said "When people are insecure, they'd rather have somebody who is strong and wrong than someone who's weak and right."
McCain really has nothing else going for him except the chance that enough people are too racist to vote for a black man, no matter how attractive and right, and they will either stay home, or vote for McCain. His campaign ads, with their coded racism, are designed to encourage this.
But, if McCain and Cheney can engineer a cold war with Russia that begins to get intense around October, why we have something much more effective than an "October Surprise." We will have the re-emergence of fear of the Russian Bear, the ten-foot tall bogeyman who was so effective in his earlier Soviet incarnation. We can go right back to 1949, and many old people would feel right at home--and vote the right way. Even younger people in the US are likely to respond to a Cheney-McCain revival of the Cold War with the Russians: vote for the guy who's a military man.
Bush foreign policy towards Eastern Europe has been traveling towards something like this for some time. The push to sign up allies among former Soviet states, while excluding Russia from NATO was one thrust; it was supplemented by the push to set up an anti-missile system (breaking the ABM treaty with the USSR) and building radar and missile batteries in former East Bloc states like Poland and the Czech Republic. The administration claimed that the system could be set up with Russian participation, and would not target Russia. Russians don't buy that: a Russian general just warned Poland that it was making itself a prime Russian target by signing up for the US anti-missile system. Critics of Bush's Eastern Europe policy have warned for years that a hostile Russia would be the result. It may have been the intended result on the part of people like Cheney and the people behind McCain.