My husband does the despatching, and he has used the same method for both ducks and geese.
Take your fowl by the legs and hold it head down. Lay its head on the ground - it will automatically bend its neck back so its head lies flat. Lay a strong stick (broomhandle is ideal) across the back of its neck and stand on the stick on each side of the bird's head. Then pull the bird's body sharply upwards. This dislocates the bird's neck quickly and efficiently.
There will be some internal bleeding, so hang the bird head-down to make sure the blood collects in the neck cavity rather than the body. The advantage of this method is that it is not in the least messy, and it is definitely quick.