Tire balance is a good idea, but that would have continued even when I put the clutch in, right? So I don't think that's it.
About the motor mounts, I took him to mean that the rubber was missing, but I'll check that. Maybe the mounts were just real worn. I don't think it was tune-up because before the vibration started there was no sign of any vibration and after I heard it start it never stopped. I hope it isn't the rear end or transmission.
Thanks to both of you for the ideas.