Well, brand name, as well as 'famous cheese', doesn't mean 'good cheese',. If you go to big supermarkets or very-visited restaurants, you will not find good cheese (many customers => larger quantities => lower quality => higher profit). You must have the access to smaller cities, less-famous restaurants and know what to eat where and when (out of the mass production and consumption chain).

There are many types of cheese in Italy, some of which can be only found in certain regions.
I'm sure you know about mozzarella, but if you came all the way to Italy, you would eat the best mozzarella, the mozzarella of 'bufala campana', that can be found almost everywhere.

In the northern Italy it's produced cheese with more natural, softer flavor; instead, in the southern Italy the cheese is seasoned longer, so it's flavor is stronger.

I suggest you, if you want to go in search for cheese, to go to the northern Italy because of the facts that were discovered by the police about cheese in some dairy plants in the south (expired, rotten cheese was recycled in making new cheese; or water was added to milk).
Near where I live the are places like the cities of Branzi and Gorgonzola , and the Taleggio Valley, that produce homonymous types of cheese, and here nothing like that has been heard.

Greetings from Italy

Italian cheeses are made in a particular locality of Italy and each has its own distinctive flavor. Here are my recommendations:

· Bra (Tenero and Duro)
· Canestrato Pugliese
· Fontina Val D'Aosta
· Gorgonzola
· Grana Padano
· Mozzarella di Bufala Campana
· Parmigiano-Reggiano
· Pecorino Sardo
· Pecorino Toscano
· Piave
· Ragusano
· Ricotta Salata
· Robiolo di Roccaverano
· Taleggio

Source: http://www.gourmetcheesedetective.com/Italian-cheeses.html