The Spline on Surface command is the way to do this. I was looking for a similar solution to the original poster; to sketch a 3D sketch onto an organic (was generated with a Loft feature) surface.
Spline on Surface is extremely powerful, it can be a bit
tricky putting relations on them though so you might need to
manually tweak them until you're happy with them. But that's a
small price to pay for its power.
Also, if you click on endpoints of a fairly curved face, your
spline on surface might not be created - in that case try placing
an additional point somewhere in the middle of the surface.
Depending on the complexity I have found splines might need more
points to pass through to successfully build on surface.