While I really don't have a problem with the way it is now, I fear that with all the murmurings around the league it will have to change soon, and I'm dreading seeing something like what college football has. If the NFL adopted an overtime system like CFB, it might be enough to turn me away from the NFL for good.
The problem, as many people see it, is that the team that wins the coin toss has an unfair advantage over the other team, which, of course, is ridiculous if you believe in the axiom "defense wins championships" which I don't, but nevertheless don't mind seeing it proven. And I don't mind letting the kickoff ride on the flip of a coin. I am a Cormac McCarthy fan after all, and both teams got there the same way the coin did. Furthermore, a good team should be able to rely on either their offense or defense when called upon to do so, and you don't always get to decide which.
What I don't want to see is a game determined differently in overtime than it is in regulation. The CFB system eliminates many things that are vital throughout the course of a game, like field position, long, sustained drives, kicking, long passes, and turnovers. Not to mention the fact that a game which ends 10-10 in regulation should never be 52-49 after overtime.
So here's my solution: the team who last held a lead in regulation gets to receive the kickoff in overtime. This way everybody knows what's coming, and what is at stake when a team scores a game-tying touchdown at the end of regulation. Will the coach decide to go for two and win it here, or will he take his chances on defense? How about kicking a game-tying field goal from the 10 yard-line with one second left when you know your defense hasn't stopped the other team all day? Are you better off taking another shot at the end zone? This solution will serve to add more strategy to the game and thus, make it more exciting and eliminate any complaints about fairness. May it be so.