A few years ago, I had my phone ripped off my arm a block from my apartment (as in where I live in Brooklyn - NOT touring around overseas!) when I was starting out for a jog. So...
Never had an issue in Florence for a week, in Barcelona for a few days, or in any stop on my 12-night Med cruise two years ago (Villefranche, Livorno, Rome, Amalfi Coast, Venice, Ravenna, or Kotor). As
@Belle2002 said, be aware of your surroundings and as
@olafLover said, trust your instincts. Walk with confidence. Odds are you'll be fine!
One thing I've found to beat the "Look at me, I'm a tourist!" whipping out of a map and/or tour book is before I leave, I figure out where I am going to want to go in terms of major sites (which I'd hope you're doing anyway), then go to Google Maps and print out walking directions from point to point, each on a separate paper. Example: On my day in Rome, I knew that the bus was dropping us off at the Vatican. I knew I wanted to go to the Spanish Steps, the Trevi Fountain, and the Pantheon where I was meeting a friend and we were going to a restaurant I learned about that has their entire menu duplicated gluten free ("senza gluten in Italian), and then back to the Vatican. I had a sheet of letter-sized paper for each point-to-point that I could check, so it was not as obvious I was a tourist as if I'd had the huge tourist map.