Those passport/wallet holder money belt things are a target for professional thieves. They know where to look and you've labeled yourself as a tourist with a taunting signpost saying "THIS IS WHERE MY IMPORTANT THINGS ARE"!
There is a happy balance between keeping your money hidden enough to deter thieves, but accessible enough that you don't need to perform a magic trick to get your wallet out for an espresso.
Now, if you really want to wear a money belt, I'm not going to stop you. After all, I don't want you to come back to me saying it's my fault if you get robbed, but I'd suggest trying a few of these tricks first.
This should get you started. Above all things, when it comes to staying safe, common sense goes a long way.
That said, don't be scared! Get out there!
Do you have any clever tricks for keeping your cash safe when traveling abroad?
Have you been robbed or pick-pocketed? What did it teach you?
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