I am a falconer so have a bit of experience here
Is the chicken being plucked first? Large amount of feathers around?
There are many different birds of prey and other predators that could be killing them, if only the breast is being removed and you are only losing one bird per week then it sounds like quite a small predator, stoats/weasel around?
It's obviously too small to move the kill to a safe place.
a buzzard would probably eat a fairly large amount and if you left the carcass out it would probably clear it in 2 or 3 days.
If it is a bird of prey you either need to remove the food source or get a larger predator around, dogs are good, and a large plastic owl etc may work.