Depending on your purpose and the number of requests you need to put through the service, the Bing Maps REST APIs are generally free to use - they just require you to sign up for a key first from https://www.bingmapsportal.com/.
The API for the Route service is at:
http://msdn.microsoft.com/en-us/library/ff701717.aspx, which can return driving distances and routes plotted between up to 25 waypoints.
Conditions for use are at http://www.microsoft.com/maps/product/terms.html, but it's basically free for educational/non-profit use, and free for commercial use so long as it's a publicly available app and you don't exceed 500,000 transactions in a 12 month period. Otherwise you need to get a commercial licence.