The small fringe shifts observed in the Michelson-Morley ( MM ) experiments are dismissed by mainstream physicists as experimental artifacts. However, this interpretation is increasingly being challenged. Based on a new insight, simple and modified MM experiments are proposed that should give much larger fringe shifts, typically more than sixty fringe shifts. The mystery to increase the sensitivity of the conventional MM experiment is to slightly adjust the angular positions and the distances of the beam splitter and/or the mirrors so that the angle between the longitudinal light beam and the transverse light beam at the source is as large as possible, say 10-2 radians. This angle is very small in conventional MM experiments, making the experiments insensitive to absolute motion. Conventional theories such as ether theory and special relativity consider a single beam up to the beam splitter. The new theory, called Apparent Source Theory, has the potential to consistently explain many of the previously enigmatic and controversial light speed experiments. However, getting a non-mainstream theory accepted is a major challenge in physics today. The only way to possibly convince the scientific community is to test and confirm a unique prediction of the theory.