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I do not have a 5D3, but since no one has replied I'll at least tell you what I use on a 5D2. I may mention a bit more than just AF settings.
I generally, but not quite always, have the camera in burst mode. However, most often I don't actually burst, but instead do a light single press to make a single photograph. If I need to burst I simply leave my finger on the shutter longer.
Most often I use AI Servo mode. If the background is plain - e.g. sky - I'll likely leave all AF points on so that I can place the bird (or birds) in more areas of the frame and still get AF. However, if the background is complex, I'll probably use only the center point in order avoid having the camera AF on something behind the bird, even though this means I have to center the bird in the frame, thus reducing compositional flexibility.
There are variations depending on the situation. In a few cases I may not use AI Servo, using single shot mode instead. I do this if I want to have that added flexibility of bird placement in the frame. I can, sometimes, AF the bird in the center and then move the camera a bit to place it off center when that is a better composition. (And, yes, I can sometimes think about composition while tracking a bird. I don't always succeed.)
With some birds that were following a predictable path to and past me, I have even used the technique of pre-focusing and turning AF off a few times.
I raise ISO as necessary to keep a high enough shutter speed, if minimizing motion blur is what I'm after. (Though I also do some shots in which I intentionally shoot at lower shutter speeds in order to use motion blur.) If possible, I use a slightly smaller aperture since this makes focusing a bit more forgiving. However, quite a bit of such shooting is done in less than optimal light, so that isn't always possible.
One more thing. I'm convinced that success in this sort of shooting depends as much or more on practice as it does on "the right" settings. You can have the best AF system in the world, but the skill of keeping the AF points over the moving target requires a lot of trial and error and practice.