In the search for rationally assembled functional materials, superatomic crystals (SACs) have recently emerged as a unique class of compounds that combine programmable nanoscale building blocks and atomic precision. As such, they bridge traditional semiconductors, molecular solids, and nanocrystal arrays by combining their most attractive features. Here, we report the first study of thermal transport in SACs, a critical step towards their deployment as electronic, thermoelectric, and phononic materials. Using frequency domain thermoreflectance (FDTR), we measure thermal conductivity in two series of SACs: the unary compounds Co6E8(PEt3)6 (E = S, Se, Te) and the binary compounds [Co6E8(PEt3)6][C60]2. We find that phonons that emerge from the periodicity of the superstructures contribute to thermal transport. We also demonstrate a transformation from amorphous to crystalline thermal transport behaviour through manipulation of the vibrational landscape and orientational order of the superatoms. The structural control of orientational order enabled by the atomic precision of SACs expands the conceptual design space for thermal science.