The goal of this study is to determine how bulk vibrational properties and interfacial structure affect thermal transport at interfaces in wide band gap semiconductor systems. Time-domain thermoreflectance measurements of thermal conductance G are reported for interfaces between nitride metals and group IV (diamond, SiC, Si, and Ge) and group III-V (AlN, GaN, and cubic BN) materials. Group IV and group III-V semiconductors have systematic differences in vibrational properties. Similarly, HfN and TiN are also vibrationally distinct from each other. Therefore, comparing G of interfaces formed from these materials provides a systematic test of how vibrational similarity between two materials affects interfacial transport. For HfN interfaces, we observe conductances between 140 and 300 MW m^-2 K^-1, whereas conductances between 200 and 800 MW m^-2 K^-1 are observed for TiN interfaces. TiN forms exceptionally conductive interfaces with GaN, AlN, and diamond, that is, G > 400 MW m^-2 K^-1. Surprisingly, interfaces formed between vibrationally similar and dissimilar materials are similarly conductive. Thus, vibrational similarity between two materials is not a necessary requirement for high G. Instead, the time-domain thermoreflectance experiment (TDTR) data, an analysis of bulk vibrational properties, and transmission electron microscopy (TEM) suggest that G depends on two other material properties, namely, the bulk phonon properties of the vibrationally softer of the two materials and the interfacial structure. To determine how G depends on interfacial structure, TDTR and TEM measurements were conducted on a series of TiN/AlN samples prepared in different ways. Interfacial disorder at a TiN/AlN interface adds a thermal resistance equivalent to ∼1 nm of amorphous material. Our findings improve fundamental understanding of what material properties are most important for thermally conductive interfaces. They also provide benchmarks for the thermal conductance of interfaces with wide band gap semiconductors.