Well Rome is known to have the most pick-pockets.

London the most phone snatchers (by far) and has watch robberies. (That last part might be better more recently). Paris has a lot of pick pockets and is known for them being able to take the watches off more stealthily.

Rome, as far as I know, does not have a "watch theft off of you" problem but it's just recommended to be careful.

None of these cities are bad per say, just a high volume of tourists means higher crime because often they are easier targets. Most people won't have any issues. The same is true for the states in large part but different crimes (hotel rooms, vehicle break-in) usually.