Yes - unless everything is balanced/just right, you will get 'some' vibration in the driveline. ( - part of my day job is driveline analysis)
If U-joints are worn, you might also hear or feel a 'clunk' whenever you suddenly accelerate or let off the gas. Just get underneath and check them.
If the motor or trans mounts are worn, whatever natural vibration there is will be worse. Again, just take a look. If they are mashed or cracked and split, that might part of the problem.
If your rear springs are worn/sagging, you could be getting some axle wind-up, which would throw off the alignment and give you some pretty strong driveline vibrations.
You didn't mention any suspension mods, so assuming everything stock. Raise the suspension and things get worse.