We have 2 LGD's, a neutered male pyr and a spayed female Anatolian. We got them primarily to guard our Nigerian Dwarf goats, but we also have chickens, guineas, & ducks so I can give you my experience with that.
We started with the Pyr in a small pasture with the goats (60ft by 120ft) and the chickens free ranging on the rest of the property. The pyr did great protecting the goats but killed any chickens that made their way into the goat pen. Now, this was when he was still in his puppy phase (about 1.5yrs-2 yrs) which is pretty typical. We tried a few ways to train him out of this, some were well-planned and others were...less so. (I will admit that I once chased him around the pasture with a dead chicken, screaming and crying at him....but to be fair it was one of my favorite hens and I was pretty devastated to find her.)
Not sure if any of the training methods had an impact, more likely the combination of him getting older/more mature and him being able to see the birds nearby and make the connection that they were part of the farm and part of his guarding responsibility. Now we have the whole 5 acres fenced and cross-fenced so that the goats & LGDs are on a big section that makes an 'L' shape around the house and the chickens free-range in the acre or so around the house. We have several chickens who like to wander in and out of the goat pasture at will and the LGDs leave them alone even if the birds wander right next to them.
Though their main job is guarding the goats, the dogs definitely help protect the birds, since any predator would have to go through the goat pasture to get to the chickens unless they came accross the street in front of the house and over the front fence. Also, the dogs patrol the property, marking it and barking to ward off predators so most stay far away. At night, we often open the gate between the back pasture and the area around the house and coop so that the dogs can patrol there as well and mark their territory to keep away raccoons, skunks, etc. They even watch the sky for hawks and make a big fuss when any try to fly over.
Obviously, no guardian is going to be perfect, but I think ours are worth their weight in gold as we used to lose 1 or 2 birds a week at times when coyotes and other predators would come right up and steal them with impunity.