I live in Rome. I've found that for the most part, finding enjoyable running routes is much more difficult in Italy than most other places I've been. Often the roads are windy and / or narrow, sidewalks are narrow, uneven, nonexistent or clogged by tourists, and drivers on the roads are - ahem - aggressive.
For Rome, I agree with the suggestion to run along the Tiber River, or else run around Villa Borghese. You likely wouldn't be near it as a tourist, but Villa Pamphili is a far nicer choice for running in a park, and Parco degli Acquedotti is also fairly nice (entirely flat, but containing some really cool old aqueducts, and an exposed section of old Roman road. Trying to run on roads in central Rome isn't impossible (I've seen large running groups do it), but it would not be stress-free...
Amalfi area is very hilly, with narrow, windy roads. I wouldn't dare run along the coastal highway that connects the various towns, for fear of dying (perhaps 5 times out of ten you might have a near-death encounter with an Italian driver). I have stayed up on the hillside for a couple of weeks in Ravello and run down the road toward the coast, then back up again. If you are alert and careful, you can avoid being killed by a car, but may need to fight off stray dogs. And it's out and back, all uphill or downhill (depending on whether you start from the coast or up above). Other roads from the coast inland are likely similar.
Naples is a $hithole. I advise just not going there at all, let alone running there. Full stop. Do try to visit Vesuvius, though, and also Pompeii. Thinking some more, let me revise my first statement. Naples is the world's arsehole. I have no idea how people live with themselves in that place among the garbage they freely throw all over the place.
People have already told you about Venice. In the town, running is nearly impossible. You can do laps around the main square when it's not flooded. Better bet is to stay somewhere with a treadmill and grit your teeth. Great place to visit though. If you like music, I recommend wandering into some of the numerous music recitals. Even if you don't like classical or chamber music, it's a world class experience to hear people play in the bzillion little churches that have unimaginably great acoustics.
Oh and hey, have fun!