Best thing is to try not to look like a tourist and an AMERICAN tourist to boot..

(shorts & sandals in a city are a HUGE no-nos in most European cities while wearing white socks & sports shoes and jeans with the shirt tucked in and a baseball cap w/ a Washington Redskins logo.. will get you....noticed.....

) ..
Even worse try not to look like a RICH American tourist..

..(hint: people getting off in the NAPLES from their luxurious cruise ships and flashing huge ROLEX watches & jewelerry to the locals in the poorest city of Italy must be crazy....)
But since no matter what you do you will DEFINITELY look like a tourist...

...try the flg:
best thing you can do : ditch the wallet (no wallet-no pickpocketing..)
Leave your passport in the hotel safe (in some countries carrying a passport w/ you is mandatory but a cerified copy of the original is ok too) and carry your credit card in a shirt pocket or in a different pocket than the one carrying most of your money (you will ofcourse "spread" your cash between the various pockets)
Never carry large amounts of cash w/ you,use credit cards (VISA-MASTERCARD are the best ,AMEX is not accepted everywhere) for restaurants and shopping and use ATMs for currency (the exchange rates are far better than the ones you will get in the US)
Avoid large crowds of ROMAs ,you can easily spot them around metro exits and in busy squares,of course don't even think about "chating" w/ them or giving them charity ,the moment you do it is the moment you will loose all your cash (these guys are MASTERS)
Be very carefull if you use any form of public transportation during your ARRIVAL & DEPARTURE..especially if you have lots of luggage (so if you can do what most Europeans do..TRAVEL LIGHT) since you will be tired and distracted (in Athens 90% of pickpocketing happens in the metro between the harbor and the city and between the city and the airport) and an easy target (especially in a crowded bus/train)
Oh and LOOSE THE FANNY PACK mister...
Btw in reality pickpocketing in Europe (and Italy) is no worse than pickpocketing in the rest of the world.Add to this the fact that MUGGING (something way worse than pickpocketing) is almost unheard of in the Southern European countries (Spain,Greece,Italy ) & stop worrying too much!