Cars are much cheaper to rent than motorcycles. A nice car like an Audi A4 can be rented for 10 days, unlimited mileage, for less than half of what it costs to rent a motorcycle. It pays to shop around for the best deal. All the major American rental agencies rent in Europe. Deals abound on the Internet, through discount brokerages, travel clubs, etc.
Remember you are going to be riding in the mountains where the roads are narrow and the turns are tight. Therefore, a smaller car is probably more suitable. The Audi A4 that recently followed us around had no problem going over all the roads the motorcycles traveled on, even the little one-laners.