In my experience the streets of Rome (and Italy in general) are (still) very safe. I'm a woman and walk around with several thief-magnets: designer handbag/backpack, gold watch, jewellery, DSLR camera (almost always), latest iPhone etc. and I feel quite safe. Same goes for my friends. I go to Italy maybe six times a year.

I hardly ever use public transportation though. And I do think you need to be careful because given the huge amounts of tourists there will always be pickpockets. But unlike in many other countries police presence is good and the police are super nice.