Of course when Michelson said this, there were just a couple of loose ends to be figured out...the "UV Catastrophe" associated with the discrepancy between the purely electromagnetic theory of blackbody radiation, and the strange threshold behavior associated with the photoelectric effect.
Right now, we keep on building bigger and bigger colliders and can't really find anything beyond the Standard Model. It seems that the biggest advances these days are coming from Astrophysics rather than High Energy Physics. Today, the two pesky loose ends that are likely to change everything are dark matter and dark energy. What we need is a theory that explains these phenomena and an experiment to test the theory.