Motivated by connections to intersection homology of toric morphisms, the motivic monodromy conjecture, and a question of Stanley, we study the structure of triangulations of simplices whose local h-polynomial vanishes. As a first step, we identify a class of refinements that preserve the local h-polynomial. In dimensions 2 and 3, we show that all triangulations with vanishing local h-polynomial are obtained from one or two simple examples by a sequence of such refinements. In higher dimensions, we prove some partial results and give further examples.