Any "safe" killing of animals ought to be done with blunt attacks to keep the skin of the animal in as good a condition as possible. You can use blunt attacks with most (all?) weapons. It has the additional advantage of requiring more attacks than the normal sharp attacks, and thus gives you more chances to increase the weapon skill (and kicking trains the Unarmed skill).
You should aim all attacks at the head, as hitting the head is the only hit location that doesn't damage the skin (the neck does damage the skin).
If you're dealing with an animal that fights back you should obviously focus on minimizing the risks (I throw rocks at trapped predators, as that's both blunt attacks and out of the reach of the animal), and when attacking an animal that is neither trapped nor exhausted you may want to focus on actually harming the animal sufficiently that you'll be able to catch it, rather than preserving the skin on an animal that may get away.
However, as far as I'm concerned, production of bird leather is mainly a way to train the hide working skill, rather than a profitable action in itself (the leather can be used, of course, so it's not a wasted effort).