Depending on the cost of the rental, you may be required to pay a security deposit either in cash upon arrival or by PayPal. We try and avoid ones where we have to meet someone on departing day as our flights are usually early and would prefer not to have to take later ones just so a property manager or owner could meet us at 9 am to tell us we took good care of the place.
We leave our rentals in as good or better condition and is a reason why owners remember us. Being quick to pay helps too. Politeness goes a long way -- Common sense stuff. Apartments that allow for us to leave the keys in it or in the unit mailbox upon our departure works best. I realize it comes down to trust. And that goes both ways.
The initial emails will either make or break the potential reservation. If it takes someone days to get back to me (without explanation), then I wonder how serious they are about doing business. If they are very "casual" with their language, then I wonder if they are also very casual with how other things are dealt with ie. meeting time, returning deposits etc.
Places with well equipped kitchens and bathrooms are the best. I would prefer not to have to replenish dish washing soap, cleaning supplies, garbage bags, salt and pepper etc. Listings that state they will provide the "little" things we tend to have in our homes go a long way in making someone's vacation easier.
One reason we invested in a world phone is so that we could get in contact with the meet and great person upon arrival at the destination airport. They like us to call when we are within 30 min - 1 hr for timing purposes.
European pay phones may not be the easiest to use. Some won't take money anymore and looking for a payphone card can be a nightmare (for example, they only sell them at Tabacs in France). Plus first time visitors don't tend to have Euro coins. And if the automatic computer operator doesn't speak English, as I found out in Reykjavik, you are pretty stuck.
The trick with using such a phone is remembering those + signs and country codes and when you'll need them (You don't need them in Iceland. I forgot. That's why I resorted to the payphone in frustration...). It also took me a while to remember how to call home (+ sign needed for Canada)!
Below are pictures of the apartment I rented in Iceland. Makes me want to return right now.
***FYI: I've gone back and increased photo sizes to make viewing easier and hopefully more enjoyable -- Hawaii, Munich, Norway, Barcelona and Italy etc to present.***