By comparing measurements made in different directions and looking for an effect due to the Earth's orbital speed, their experiment famously produced a null result.
The theory was quickly replaced by special relativity, which gave similar formulas without the existence of an unobservable aether.
More precisely, provided that no phenomenon violates the principle of relativity of motion, there is no means to measure the velocity of an inertial observer with regard to a possible medium of propagation of quantum waves.
Einstein gives the following example: suppose two equally-composed elastic bodies are in space and distant from each other such that the interaction between them can be ignored, and whose only relative motion is a uniform rigid rotation around the line joining the centers of both bodies (like spinning wheels around an axle).
The non-rotating-spheroid frame has physics whose cause lies outside the system, responsible for the oblateness of the spheroid.