My old dog was a staghound x. 'nuff said! I brough her home from the RSPCA totally oblivous of the problems I could've had with the FOUR cats living at our place. Three belonged to my landlady that I shared the house with, the other one was mine: a big ginger tomcat.
Let the dog into the backyard and off the lead (really, first dog, no idea!), dog saw my cat, gave chase, indignant cat turned around, puffed himself up, gave her a big wack across the nose which left a bleeding scratch. This repeated itself about 3 times after that and she never tried to chase him again. She still chased 2 of the other cats because they kept running away from her, but they lived on the roof most of the time anyway. She as always fine with cats in my house or in friends' houses after that. But if my tomcat wouldn't have taught her some manners, I would've had a lot of trouble teaching her not to chase.
My current dog's prey drive is nowhere near as strong as that. But she still loves chasing our 2 foster kittens. They do run to a safe place, but as soon as the dog stops, they will walk straight up to her and rub themselves against her. I believe they spent their first months in a household with a few dogs and other cats and nothing phases them. I notice my dog is getting less motivated to chase them now. This morning I hid a couple of chicken necks in the backyard and then I watched the dog helplessly looking on as the cats stole one from under her nose and ate it. Normally I would've interfered, but I was in a real hurry and she had another neck and fresh bone hidden in the yard. It was damn funny to watch though.
Just to say, it very much depends on the cats too! They are very resilient creatures. And very pliable when you get them as kittens.