Here are my thoughts on risks and safety for an American in Rome 

Your biggests risks are associated with the following:

Drinking too much, which makes you vulnerable to others (and makes you more likely to make stupid decisions)

Running in or near traffic: Roman drivers are used to avoiding walking pedestrians, not to people who run in traffic

Being in crowds or crowded places (which makes pick-pocketing much easier)

Wandering into bad neighborhoods, especially at night

Consequently,

Don't drink too much

Waddle across streets; don't run

See the city in groups, not all by yourself

Be especially careful of you wallets and purses when you are on buses or subways, in the train station, or in crowds of any sort

     The #64 bus is notorious for theft; theives may work together, or even in groups

     Do not keep your wallet out and "wave money." If you are making a purchase in a crowd, be quick and careful about it! 

Stay in the city center; learn to read a map; remember landmarks for getting home; have money for a cab if you need one