There's a general, informally observed pattern across the country worth knowing rather than treating as a hard rule: northern cities and mountain regions — Milan, Turin, Bolzano, and the Dolomites in particular — tend to have the most consistently dog-accommodating culture, with a dense concentration of dog-friendly cafés.
Southern and more rural areas can be a bit more traditional about pets indoors, though outdoor seating is nearly always an option regardless of region, and attitudes are shifting as pet ownership rises nationally. A polite question at the door, as covered on the Restaurant Etiquette page, smooths this over anywhere in the country — it's rarely a hard no, just a case of asking rather than assuming.