If I use metal conduit, do I need to run a 3rd green wire for ground?

I suggest that you do run the insulated grounding wire.

Connection points along the conduit run might loosen, come apart or rust through.

The insulated grounding wire will eliminate any of those problems.

Is this new construction or a remodel? Trying to use the conduit for a ground is not code and can be dangerous. Where do you live? Is cable allowed, if so cheaper than conduit and pulling 3 wires.