I know what you mean about Rome and some of the problems. I traveled to Europe quite a bit on business and very frequently to Paris and London and I was careful but not overly so. I also worked in lower Manhattan for many years and also near 57th Street and never had a problem as I made sure to just be sensible. It makes good sense to make copies of your passports before you go and keep one copy away from where you keep your original and also leave one copy home just in case there is a problem. You can always have it sent over in a hurry. I know in Rome there is an exposure of a number of children running up and distracting you while one takes the wallet, purse, etc. Some of these are gypsy children and they are good at what they do. I subscribe to act like you would in NY City. Be aware of the surrounding and if something does not look good it probably isn't.
Unfortunately a lot of the note is about problems when it fact it is terrific to see the cities and enjoy them and the culture. A year and a half ago my wife and I rented a car in Paris and drove through France and Italy to Tuscany where we had rented a villa and just enjoy the European food and country. We never had a problem so I am reluctant to put too much emphasis on the problems and ignore the beauty and experience of Europe.
Have a great trip..!!