OSRM gives you routes with road geometries as they are in the OpenStreetMap database. For example, you can get the route as GPX (and post-process this file if you want). This would look like the following:
GET http://router.project-osrm.org/viaroute?hl=en&loc=47.064970,15.458470&loc=47.071100,15.476760&output=gpx
Read more: OSRM API docs.