I've had 3 greyhounds & a lurcher, 2 very high prey drive and other 2 would take chase as its fun but were more distractable. Most of neighbours surrounding us & on the street had cats and a few would come in the garden.
If garden is secure its fairly easily managed if a bit of a pain, just take dog out on lead into the garden and check its clear of cats before letting dog loose. I realised looking g out window wasn't sufficient when the cat is behind the door
Some cats did get used to the sound of door to make a run for it , but a few will then sit just out of reach to taunt, which then can over arouse hound & practicing the behaviour you don't want.
My greyhounds didn't spend long enough outside (5 min run & back to bed) to keep the cats out of the garden, it was only when mutt hound arrived who liked to be out hours & regularly patrol they gave it more of a wide berth.
Never really had an issue walking in the neighborhood, used to take 3 out together.
When high drive girl was new she'd rear, spin & try to grab the lead to get to a cat casually sat on a drive not giving 2 fucks while I wished it would just move out of sight. Over time it just became an excited dance as realized it was pointless.
Best to make use of a muzzle early on while learning their reactions, you can usually tell by the ears when interest is piqued or spot something before them, they aren't always that observant unless its moving. A muzzle will make you transfer less anxiety along the lead & usually already trained. If associate it with nice things like walks & treats its just like wearing a collar or harness to them, its useful in many occasions so I trained our mutt early on to happily wear one. Mine didn't go off lead in public but were easy to tire out with sniffy walks & are lots of secure fields/indoor schools can hire if need somewhere safe to run. Even when racing they have trot paced walks daily for maintaining condition.
I actually stayed in a b&b for a weekend with the 3, (2 high prey drive girls) that the owner assured me could separate from their cats from so be fine.. Only they didn't bother, luckily only soft boy who discovered a hissing cat & the girls were blissfully unaware of its existence. After that we checked hall was cat free before going out with the muzzled hounds.