Will update as I learn more. Veterinary research is always on going.
the family SHIGHTHOUNDS (that Italian Geryhound are the smallest) are not like other breeds, here is  a MUST READ. in fact print it and give it to your vet  Are sighthounds really dogs?

Good breeders breed for  3 things

1,temperament
2,health
3,beauty (does it look like the breed?)

disclaimer I am not a vet. please do you own research as well.

there are many health tests that can be done. some are reliable like DNA genetic testing. some are not so much. some only good for that day and not 6 months or even a few weeks later. some not worth doing if the dog isn't showing any signs and some blood tests have such a rang of "normal" that you don't get a clear picture. while some results are up to the opinion of the vet and may differ with that of another vet.

Main health concerns in IGs

Legs  addressed in Is this really the breed for you? page

DNA testing
most are breed specific. meaning the tests were studied and reliable  ONLY for the breed it is labeled for. so what maybe,lets say  reliable results  for a beagle, may not be for a chihuahua.

For IGs

Teeth- besides the issues with teeth common in dogs and more so in toy breeds, autosomal recessive amelogenesis imperfecta (ARAI), also known as familial enamel hypoplasia (FEH), is an inherited genetic disorder of tooth enamel that occurs in Italian Greyhounds.

thankfully we do have DNA testing for enamel hypoplasia
so a breeder can reliably breed pups that will not be affected (not have the problem)


PRA
Progressive retinal atrophy is characterized by progressive degeneration of the photoreceptors in the retina that leads to blindness. On average PRA in Italian Greyhounds is diagnosed at 6.5 years of age.  https://vgl.ucdavis.edu/test/pra-susceptibility-ig

thankfully we do have DNA testing for PRA IG-1 A,B,C,D and E.
so a breeder can reliably breed pups that will be low risk/not affected.


  PCAG
Primary closed angle glaucoma is a hereditary disease that results from a rapid build-up of pressure in the eye, which causes loss of optic nerve function and retinal ganglion cell loss. The onset of this condition is sudden, painful, and if left untreated will lead to blindness. The average age of dogs at diagnosis is 3.9 years.

Thankful we have a DNA test for PCAG BUT they can only say that Clear(no gene markers for it) is normal/low risk (2%)

Cryptorchidism or Retained Testicles
Very intersecting read.  by Cheryl Lopate, MS, DVMDiplomate, American College of Theriogenologists
Cryptorchidism in the Dog How it happens, How to diagnose, Whether to Treat

thankfully this is something we can see so no test needed.


Color Dilute Alopecia

This is were the  colored hair breaks off at the skin. uncommon in igs but can happen in diluted colors. being blue,blue fawn, fawns without black nose leather.

though this can happen in any bloodline with dilutes. many feel (i,myself included) that there is a stronger chance of this happening when a dilute colored dog is bred to other dilute colored dog. so i do not do this.


Seizures and idiopathic epilepsy

This is a tricky one

As I read more I find that it seems that there are thoughts that sight hounds are prone to seizures. And many thoughts on why this is.


Yes they are sensitive to toxins ,over medication, react to flea/tick meds, vaccines…sometime food.. some plants...household cleaning products....

most think it is due to their strong hunting instinct. Being a family of dogs that hunt by sight they are  “hardwired” to notice EVERYTHING and could be sensory overwhelmed.


 while I fully support the removal of affected individuals and the practice of not repeating the pairs of parents of known affected pups. The breeders dilemma is that currently there is no DNA test and though some seizures are most likely a defected gene ( or such as toxins, brain tumor, tick illness....) the mode of inheritance (how it is passed on) is unknown (does both sir and dam needing to be gene carriers? only one needs to be? a combo of genes needed? good read ) AND it may differ in each breed. Meaning no matter what one might do, until there is DNA test, specific for that breed. it is all a genetic lottery tossup and always a risk. No line of dogs or breed (or mix) can say that there is no chance of producing offspring with it. 

So with that, a breeder must weigh the risk of cutting out whole branches of family tree. Permanently removing gene pool that have already been DNA tested and have passed what health tests that we CAN definitely know (issues at effect the dog's day to day life with blindness and  pain) for the risk of issues that has not been seen in the dogs that would be removed. Simply, seizures are something a breeder can not guarantee against at this time but there is hope.


"In summary, the complex nature of canine epilepsy will prove challenging with respect to determining the underlying genetic mechanisms responsible. However, the advancements in the DNA technologies will enable scientists to unravel regions of the chromosome regulating this debilitating disorder."

Genetics of Canine Epilepsy
Tufts' Canine and Feline Breeding and Genetics Conference, 2007
A.M. Oberbauer, PhD
Department of Animal Science, University of California, Davis
Davis, CA, USA

Luxating Patellas (slipped stifles or dislocated kneecaps)
this can show up as a skipping with the back legs or off and on reluctance to put weigh on a rear leg. some dogs will stretch and pop the kneecap back in place. it is graded 1 to 4. some needing surgery, some not. breeders can reduce chances by breeding from dogs with good angulation to the rear legs and by having the stifles/patellas( knee caps) checked. some vets, if very gently, can do this at the puppy check but it is best to wait until a  year of age for a true feel.