I don't know if it's actually feasible. As you said cats are natural hunters, and it's natural for them to stalk prey like birds. Furthermore cats are pretty hard to train.
Even if you were to miraculously train them not to hunt, there's no way you could train them to selectively not hunt only birds.
The solutions are two in my opinion.
1) keep your cat indoors. I wouldn't suggest it for outdoor cats, they have a right to live in the environment they are used to. But it's a solution, albeit drastic. It will also keep them safe from other dangers.
2) tie a little bell around their necks. The bell will be noisy when they move, making it much harder for them to stalk a prey. Still, any prey, not only birds.
