Here is another look at training ex-racing greyhounds to down and down stay. Of course, this method would work on most large dogs. I always teach ex-racers to down from a stand since they may or may not have a good sit... although I think it is really silly to have to sit your dog first and then down. They should all down from a stand.
One of the big differences in training an ex-racer versus any other dog is that they rarely respond to pressure. If you try to pull or push them into a down, greyhounds resist and lock up. It rarely works.
I also think they have to "learn to learn". Their entire lives have been about routines and good habits (which is why they are often better pets than those raised by Joe Average dog owner, in my opinion). Racing greyhounds are not really asked to do anything so responding to a cue is completely foreign.
I start teaching down by luring my greyhound under one or both legs. Depending on the height of the dog, I may sit on the ground, on a bucket, or a chair without arms (you need to be able to swivel). This is Sahara's 2nd session. He had already worked for a few minutes on a prior day so you are going to see him lie down faster than your own greyhound might. Just remember he has had a few extra minutes of practice. I am also clicker training which you do not have to do, but it does make your training more precise. My goal here is to get Sahara to lie down and then to continue to reward him for staying down for as long as he offers.
Notice that I am mostly quiet. I am not giving him any verbal cues and not distracting him with chatter. He will be taught a hand signal first anyway. Remember to reward your greyhounds for poking his head under your leg and for putting his elbows on the ground. You want to reward all of his steps in the correct direction and not just focus on the full position initially. Let me know if you have any questions.