raster - Joining 2D polylines and DEM into 3D polylines using QGIS/GRASS/SAGA?

I have a road network in 2D/PostGIS and a DEM. I want to turn the road network into 3D by retrieving the Z-value from the DEM.

Both files are quiet large so an efficient process is needed. I'd prefer to use QGIS or any other open source tool.