I have rented at least a dozen apartments in several different cities in Italy. I've yet to find one that didn't require payment in cash. They may require a credit card to reserve, but I have a feeling that cash is just the way it is, unless you are going through some major agency.

When leaving from the states you can call them and ask them to raise your daily ATM withdrawal limit. This reduces the number of withdrawals, each of which has a fee.

Unless its a cheap apartment for a week where I'm OK with walking around with enough euros to pay for it, I usually tell the owner that this is all the ATM would give me, and to come back tomorrow or the next day for the rest. I never had one say, "No, you cannot stay here unless you pay it all now." They have the choice of accepting your reasonable request, or leaving the place empty. They have always understood that I'm not going to walk around with a thousand euros in my pocket.

I've had some places request that I wire the money before hand because in their words, "You have to pay in cash, and you don't want to walk around with so much cash." So it seems to be understood. Unless I'm dealing with a major organization I don't like to wire cash to an unknown party. I always say that cash is not a problem. then I show up and tell them that this is all that I could get out of the ATM today, come back and I'll give them the rest. It's an understood practice

Once, the advertised wifi connection didn't work, and I had already paid the full amount because it wasn't a very expensive place. When I called the owner and said I needed the WiFi so that I could work, he said he was hundreds of kilometers away in Puglia for the rest of the month, and there was nothing he could do. A few days later I bumped into him in the street.

Another time i hadn't yet paid the full amount, and told the owner that this is all the ATM would give me, come back in a few days. Turns out the air conditioner didn't really work very well. In the morning I called him and said that the place is an oven, you need to fix the air conditioner. Repair men were there very quickly. So paying what you can is not a bad policy.

Bottom line, don't sweat it. They know you don't want to leave the place and be without a place to stay, and that you will pay the rest. They also know that you are not going to be able to find a customer at the last minute, and won't turn you away if you a reasonable amount up front, and the rest over the next few days. Give them what you can, and make additional trips to the ATM as needed.

Going to a U.S. bank and converting dollars to euros is like getting robbed. I wouldn't do that.