You really can't lump Athens in with Rio de Janeiro. In my 50 trips or so to Rio, I've seen every type of crime, including muggings under the canopy at the entrance to the JW Marriott. While there is petty crime in Athens, there's petty crime in all major cities. I've seen the bag theft thing many times and I just never understand why people give their bags to some random person 2 seconds after they arrive at a hotel, or leave their stuff unattended. Also, when theft victims claim that whatever was only out of their sight for a few seconds, it's almost always far longer than that.
The basic rules apply everywhere. Keep your stuff in sight, don't leave your bag/backpack/purse sitting around unattended, don't take anything outside that you don't need. Leave the expensive jewelry at home, keep the electronics to a minimum, carry the minimum amount of stuff in your wallet/purse. One credit card, one ATM card, drivers license, health insurance card, a small amount of money. That's it. As always, with basic precautions and maintaining awareness, you'll be okay. There are always easy targets around tourist areas so your job is to not be an easy target. You'll always be a target, since if you're there you're presumed to be rich, but criminals are adept at stack ranking the easiest targets. Basically, if there are four targets and you have your stuff zipped up and are watchful, and the other three have cameras and bulky purses and the women have nice jewelry and the guy is wearing an expensive watch or the kid has a $1,000 phone in his/her hands, the bad guys will hit them before you.