I drive a manual trans. 99 Celica with 74,000 miles on it. It shakes whenever I am going above 40 and accelerating. If I let off the gas a little and just maintian speed it stops.
Could be a bad cv-joint.
It could be a out of balance or damaged tyre (you might not be able to see the damage) or a suspension part or adjustment or … well lots of things.
A good alignment shop can likely spot it quickly. You can try moving the front tyres to the back and the back to the front and see if it changes. That at least will eliminate either the car or the wheels/tyres.
I forgot to mention that this has been going on for a while and during that time I have gotten brand new tires and rims.