We report measurements and modeling of thermal conductivity in periodic three-dimensional dielectric nanostructures, silicon inverse opals. Such structures represent a three-dimensional "phononic crystal" but affect heat flow instead of acoustics. Employing the Stober method, we fabricate high quality silica opal templates that on filling with amorphous silicon, etching and recrystallizing produce silicon inverse opals. The periodicities and shell thicknesses are in the range 420-900 and 18-38 nm, respectively. The thermal conductivity of inverse opal films are relatively low, ~0.6-1.4 W/mK at 300 K and arise due to macroscopic bending of heat flow lines in the structure. The corresponding material thermal conductivity is in the range 5-12 W/mK and has an anomalous ~T(1.8) dependence at low temperatures, distinct from the typical ~T(3) behavior of bulk polycrystalline silicon. Using phonon scattering theory, we show such dependence arising from coherent phonon reflections in the intergrain region. This is consistent with an unconfirmed theory proposed in 1955. The low thermal conductivity is significant for applications in photonics where they imply significant temperature rise at relatively low absorption and in thermoelectrics, where they suggest the possibility of enhancement in the figure of merit for polysilicon with optimal doping.