Private room and home rentals have always been an iffy situation, rife with everything from outright scams - people posting properties they don't own, then collecting money for rentals - to just not living up to the description. While going through a reputable broker such as White Key Villas solves some of those issues, the represented properties are usually high-end and tack on a hefty fee that goes to the broker. But there are some newer systems which present much more inexpensive but vetted properties.
How inexpensive? It starts at free and works up from there.
How safe is it? A quick glance at current offerings in Athens showed dozens of properties with dozens of reviews on each. Your appetite for risk may be different, but if you see that 35 different registered people have successfully stayed at a property, it's reasonable to assume your experience will be similar. And only registered users of Airbnb who have actually paid for and stayed at a property can post reviews, greatly cutting down on the possibility of review abuse.
Most of the owners are not professionals in the lodging industry - which may be a good thing. You will be dealing directly with a typical resident of the country you are traveling in, in this case Greece. If you do have a problem, there is not much you can do to force an immediate resolution - but this is not much different from dealing with most hotels and other travel service providers, where disputes may drag out for weeks or months after you've returned home. Airbnb properties are in the Athens area, as well as hot spots like Mykonos, Crete, and Corfu.
Despite the name "Airbnb," not all properties will actually offer breakfast. More: Airbnb