Passport in the hotel room safe. I carry a photocopy in my wallet. Wallet in the front pocket, carrying enough cash to last a couple of days plus a couple of travel-friendly credit cards (meaning no foreign transaction fees). I will usually carry my debit card as well to get more cash if I need it.
I look at it this way - if I get robbed at gun/knifepoint, I'm doing whatever the perp wants anyway. I'll go get him cash if he demands it. If it's a petty thief, I'm out a photocopy of my ID plus my cards and minimal cash. Amex Platinum will take care of getting me a new card same/next day and will advance me enough cash to cover my hotel and living expenses, if needed - or they will arrange to pay my hotel directly. My bank will probably be a bit slower getting me a new debit card, but they will shut it off and protect my funds.
I basically rely on credit cards as much as I can when traveling since they generally give the best exchange rate, especially since they got sued. ATM cards are good for converting cash at a favorable rate, but I don't usually carry a ton of it anyway. My passport is what I really need, so it doesn't stay on my person when I'm out. I don't really go anywhere too odd, so if a western european country decides my photocopy isn't good enough I'm sure we can straighten it out by walking back to my hotel. If not, we'll lose some more time but I'll call the nearest consulate. Either way, it's never actually happened to me. I have never needed to even show ID in a foreign country for anything except checking in at a hotel in Italy - and they just make a copy.