When I did my tour I carried my passport at all times. I had a money belt which had my passports, spare cash card, and emergency money which I wore when we were in sketchy places, but most of the time it was in my day pack inside a pocket.
Theft is everywhere but if you are careful then you won’t have any problems. Just don’t walk down the street holding your wallet in your hand or in your pockets. 2 people on my tour got robbed - one girl getting out of a taxi in Barcelona with her wallet in her hand, and our driver in Istanbul got pick pocketed.
Just be careful and you will be fine. And beware of the Gypsies - they are always out to get your money