Those clever experiments developed by scientists along the years, as light spectrum on a prism, resonances, beam interruption wheels, mercury barometer column, magnetic induction, Millikan's, and many other share great lucidity in their conceptions.

 

How would you design a clever experiment to measure the propagation speed of gravity ?

It has been done through the rate at which a couple of pulsars loses its orbital energy, and rewarded with a Nobel prize.

 

This experiment, where the orbital speed modulates the pulsar's emission frequency through Doppler effect, indicates some orbital energy loss through the radiation of gravity waves, and the rate of loss is consistent with gravity propagating at the speed of light. Though, you have plenty of room to improve the accuracy!

 

Older experiments involved Webber bars, but as far as I understand (=little) these experiments produced and detected only near-field waves, which don't even need propagating waves (with a finite speed) to exist.

 

A serious difficulty: gravity waves haven't been detected up to now (I believe), which won't help measure their speed.