It's unlikely that the transmission is the culprit. Auto or manual, btw?
What I would inspect, though, are the U joints on the driveshaft on both ends. Make sure they're good and solid. Surface rust is ok, but orange dust or red dust, etc is a bad thing. You can use some liquid lubricant on them if you like. Can't hurt anything.
EDIT: With the tranny in park (auto) or 1st (standard) and the vehicle on the ground, that driveshaft should be rock solid, I believe. I'll look at this tomorrow on mine and see how mine behaves, but I'm fairly certain that it should be rock solid.