There are a variety of exchange services that pair families in different countries who want to swap homes for a week or so each year.
HomeExchange.com is the graybeard in the business. If you’ve ever seen the movie The Holiday with Cameron Diaz and Kate Winslet, this is the service they were using to swap homes in Los Angeles and London.
Homes in the United States are often much nicer and more spacious than accommodations in European countries, so be realistic in your expectations if you're traveling overseas. You can also use the service for domestic swaps.
HomeExchange is a freemium service where you're offered some basic functionality with their online listings. The hope is that you'll love the service so much you'll want to pay the company to upgrade your account to a pay model.
Other sites that offer similar free or pay services include HomeLink.org, JewettStreet.com, and SabbaticalHomes.com; the latter is geared toward the academic crowd.