There are 2 ways to "brush" your greyhound's teeth: do it yourself or let the greyhound do it. I can tell you from first hand experience and observation that the people who figure out a way to let their hound brush their own teeth have a dog with consistently cleaner teeth and healthier gums.
How does your hound brush his/her own teeth? Gnawing. Regular, prolonged gnawing. Forget the Milk Bone theory (a bite and it's gone). Forget the dry food theory (a gulp and it's gone).
Find something your hound really can gnaw like rawhide (yes, rawhide - actually shown to clean tartar), raw bones (emphasis on RAW), raw turkey necks (for the brave amongst us - they are a meal for the day), many of the "body part" treats like the tracheas (moo tubes) and hard jerky treats.
When you brush this way, your dog does the work and their teeth are awesome.
Naturally, there will be exceptions... hounds with a gum disease like CUPS, hounds with major root exposure. For the majority, though, it works if you keep up with it.
Re rawhide (I know some people have an instant aversion to the word) please don't be afraid of it. You hear horror stories about "choking" but the vast majority of those stories are greatly exaggerated. It's not "choking" when a dog tries to swallow a piece of rawhide that's still too big and gags/coughs it back into his mouth and resumes chewing. Duh, dogs do that.
I've been a vet for 30 years and in that time I've seen ONE dog who genuinely choked on rawhide. It got caught in his esophagus and it took an anesthetic and an endoscope to push it into his stomach. One dog in 30 years is a pretty tiny statistic.
Some dogs simply can't digest rawhide and those dogs will vomit it back up. Those dogs shouldn't be offered rawhide, of course, and other gnawing treats should be used.
That said, you should never leave your dog alone with rawhide, raw bones, etc. I don't expect problems, but I'd never leave the house with my hounds chewing anything.
So, give it a try! I am owned by 5 greys, and 4 of them self-brush, have gorgeous teeth, and haven't had a dental since I adopted them. Number 5, my senior girl, is way too prissy to gnaw on much and she definitely needs some outside help keeping her teeth clean

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Carol