In recent years, an energetic experimental program has set quite stringent limits on a possible “non-1/r2” dependence on gravity at short length scales. This effort has been largely driven by the predictions of theories based on compactification of extra spatial dimensions. It is characteristic of many such theories that the strength and length scales of such anomalous gravity are not clearly determined from first principles. As a result, it is productive to extend the current limits the range and strength of such hypothetical interactions. As a heavy, neutral, and (almost) stable particle, the neutron provides an ideal probe for the study of such hypothetical interactions at very short range. In this work, we describe methods based on neutron interferometry which have the capability to provide improved sensitivity non-Newtonian forces down to length scales at and below an nanometer.