Dogs, particularly dogs such as scent hounds, sight hounds, and terriers, have a
very strong prey drive, and sometime that can't be trained out of them. I should know, because I have tried with some of my dogs, and failed. A prey drive is, as
dlongterry explained it, a voice in the dog's head saying "Omigosh! Lookit that thing moving! GET IT... GET IT NOW!!"
The chances of training this out of your dog varies, depending on how good a bond you have with your dog, whether or not your dog has the desire to please, and whether or not your the alpha, or "pack leader".
If your dog sees you as the "pack leader", then he would see the chickens as
yours.
Also, you should try to train him. Dachshunds are
very hard to train, I should know, since I have been trying to train three of them. They are very stubborn, and easily distracted. But, if they are trained, it gives you an advantage. For instance, if you tell your Dachshund to sit while your out in the yard with him, then he cant go attack your chickens. You see, for a dog
sit means "butt on the ground", and if they have their butt on the ground, how are they going to go attack a chicken?
If training fails, you should build a fence that restricts your dog from getting to your chickens: either make your chickens' fence stronger, maybe bury it, or build a fence for your dog. Since he is a digging dog, you should bury the lower part of the fence so that he can't easily dig out from under it.
Whatever you do, don't try to
spank him for killing your chickens. Your dog really wont learn anything from this, it will just give him a negative experience with you. You can't spank the bad behavior out of a dog.
However, if you don't want to build a fence, you could always get a shock collar, stand from afar, let him start bugging the chickens again, and give him a good shock... I find this method horribly cruel, but it can be used as a last resort, and the dog won't be able to relate the shock to you, which will ensure that he won't be afraid of you.
Hope that helps!
Hannah.