Okay... a little special relativity...
Here is an experiment which someone has done to prove part of Einstein's special relativity... You've got two clocks, one you leave on the ground, and one you fly around at very high speed. Afterwards, the clock you moved around at high speed is slightly behind the stationary clock.
So.. the closer to the speed of light an object moves, the slower time affects it.
Question... can this experiment be adapted to determine the speed of the earth as it moves around the sun and the sun around the galaxy and the galaxy around the universe?
I think that if you were to get a bunch of clock and send them off in every direction. If you were to check the times on the clocks you should see variations. If the Earth were the center of the universe and everything else moved, but not the Earth, all the clocks would be the same. Given that we think the Earth moves, we should be able to determine the direction and velocity of the movement from the relative times on the clocks.