If you're looking for an Italian breed of dog that is short haired then the Italian Greyhound would be one possible suggestion. Despite its name, there are known cases of varying coat colour amongst Italian Greyhounds, although the second most common colour after grey is actually more of a tan than a red. The other factor that may cast doubt on this being the correct breed is that Italian greyhounds don't usually grow large enough to be considered 'medium size' dogs, although both the size of the dog and the length of its hair are fairly close to what you describe.
Another possible breed you could be referring to is the Seguigo Italiano, which is a breed that falls into the 'scent-hound' family. The Seguigo can come in a short-haired variety and is of medium build, but these dogs are usually a fawn colour, sometimes with black markings.
One suggestion would be to contact the ENCI which is the Ente Nazionale della Cinofilia Italiana, a national association that has this website for English-speaking users. It may be able to give you some further assistance in searching for this mystery breed.
Alternatively, there are other local groups like the Associazone Canili Lazio, a group based in the central region of Lazio, which battles against cruelty and abandonment of dogs, and helps in re-homing them. Dropping them an email might be one way of getting some information regarding the different breeds of Italian dogs that they've come in to contact with.
