I have Minecraft PC 1.8.7 and when I begin riding in a minecart while facing North, I continue to face North throughout the ride. How do I face the direction of travel?
This is actually a feature to fix a bug that existed in previous (I think 1.6 and earlier) versions of Minecraft. In old versions players WOULD be made to face the direction of travel but it was never fully synchronized and even if your minecart went around a 90 degree turn it might only turn half that or even none at all.
On top of this players traveling by minecart would be more inconvenienced by this turning. Since the angle the player turned was never predictable and often times people would turn with their mouse anyways, the auto-turning bothered many people and was reported a good few times as a bug (or feature request to be removed).
In I believe 1.7 this feature was removed for a much smoother and more free ride. So to summarize, there is no way to do this in vanilla Minecraft without the use of commands and/or command blocks.