It depends a lot on the greyhound, I think.
We had a rescue when I was little, a beautiful dog named Anubis. She wasn't too bright, but that was okay; she had a pal, an Eskimo dog called Raven, who was smart enough for the both of them and had a sense of humor to match - not but a foot high but she could jump up from a standing start and lick my face, and often did. Even on the worst day of my life, Raven could still get me to smile.
Anubis hadn't been a winner on the track but she loved nothing more than to run, and Raven knew it. Every so often, not more than once a month or so, Raven would pick a spot at the edge of the backyard fence. There she'd dig a hole bigger than she needed, because the hole wasn't for her; it was for Anubis, who'd wriggle through and promptly light out for the horizon. Meanwhile Raven amused herself and kept us busy with fruitless attempts to catch her. She'd run straight at me till I crouched down and committed to the lunge, and then she'd dodge on a dime and turn laughing to watch me faceplant in the dirt - cats might not laugh but dogs sure do, and Raven was the laughingest dog I ever did know.
This'd keep up till Anubis had done stretching her legs, and trotted back to press her muzzle into Mom's hand by way of genuine contrition. You could see she felt bad about it, but only to the extent she knew Mom didn't like it, if only because we were concerned with the day's occasions and she worried Anubis would get hit by a truck, which was a real concern on the gravel roads around where we lived then. Hell, Mom and I spent enough time ourselves tooling around those back roads with a six-pack in a foam cooler on the bench seat between us, her getting through a couple-three cans of Bud Light while she taught me how to drive and how to know every bird we saw without most times needing to pick up the Audubon next to the cooler. But just a few beers, barely enough to get buzzed, and most of the folks around there were a fair bit less restrained than Mom - less likely to worry about whether or not they might hit a deer on the road, more so to put a .308 through it, this being close enough to roadkill to keep the fish & game wardens sweet. So I can't blame Mom worrying, but Anubis must've either kept mostly to the woods or else just led a charmed life; occasional fleas notwithstanding, she never came to the least bit of harm.
As for the running itself, Mom understood that was just what Anubis had to do every now and again, but when she came back you could see she'd rather have found a way that wouldn't put Mom out. In any case, only once she saw Anubis was satisfied would Raven let me "catch" her, mainly so she could lick my face and I'd scratch behind her ears - it wasn't that I couldn't stay mad; it was that she was the kind of dog that you couldn't get mad in the first place. Both of them were.
Mom kept Rottweilers later - as pets, not for any other reason - and those were good dogs too, smart and kind and loving, but Anubis and Raven were special. God, I loved those dogs. Every kid should be so lucky.