I’ve never had a greyhound, but quite a few people at the dog park where I take Bitz the WonderMutt have them.
Many of the greyhounds I’ve met do seem a little shy (not wary) around strangers, but if their person is around and the person doesn’t seem loud or jumpy, they’ve all allowed themselves to be petted and played with.
Bitz is a great socialization dog, and she takes it on herself to a) look out for and protect shy or small dogs when big and boisterous ones want to engage in unsolicited play and b) gently play with and persuade shy and small dogs to play with her.
I’m probably giving dogs too much mental credit, but it seems that they figure out if the 110 pound Labweiler can play with them and nobody gets hurt, maybe play in general is a good thing.
Lastly, I’ve found that the getting dogs to figure out humans can play too thing that Johnathan Chance mentioned above is very easy to do: You just act like a dog. You get down on all fours, put your arms out in front of you on the ground, lower your head and quickly swivel your neck. I have yet to meet a dog that doesn’t immediately recognize this signal.
Whether you get a greyhound or adopt a different dog, thank you for adopting one in need of a home rather than going through a breeder. There are just so many great animals who need good homes that I don’t understand paying hundreds of dollars to a breeder rather than saving one in need.