The design of thermoelectric materials led to extensive research on superlattices with a low thermal conductivity. Indeed, the thermoelectric figure of merit ZT varies with the inverse of the thermal conductivity but is directly proportional to the power factor. Unfortunately, as nanowires, superlattices cancel heat conduction in only one main direction. Moreover they often show dislocations owing to lattice mismatches, which reduces their electrical conductivity and avoids a ZT larger than unity. Self-assembly is a major epitaxial technology to design ultradense arrays of germanium quantum dots (QDs) in silicon for many promising electronic and photonic applications as quantum computing. Accurate positioning of the self-assembled QD can now be achieved with few dislocations. We theoretically demonstrate that high-density three-dimensional (3-D) arrays of self-assembled Ge QDs, with a size of only some nanometers, in a Si matrix can also show an ultra-low thermal conductivity in the three spatial directions. This property can be considered to design new CMOS-compatible thermoelectric devices. To obtain a realistic and computationally-manageable model of these nanomaterials, we simulate their thermal behavior with atomic-scale 3-D phononic crystals. A phononic-crystal period (supercell) consists of diamond-like Si cells. At each supercell center, we substitute Si atoms by Ge atoms to form a box-like nanoparticle. Since this phononic crystal is periodic, we compute its phonon dispersion curves by classical lattice dynamics. Non-periodicities can be introduced with statistical distributions. From the flat dispersion curves, we obtain very small group velocities; this reduces the thermal conductivity in our phononic crystal compared to bulk Si. However, owing to the wave-particle duality at very small scales in quantum mechanics, another reduction arises from multiple scattering of the particle-like phonons in nanoparticle clusters. At room temperature, the thermal conductivity in an example phononic crystal can be reduced by a factor of at least 165 compared to bulk Si or below 0.95 W/mK. This value, which is lower than the classical Einstein limit of single crystalline Si, is an upper limit of the thermal conductivity since we use an incoherent-scattering approach for the nanoparticles. Because of its very low thermal conductivity, we hope to obtain a much larger ZT than unity in our atomic-scale 3-D phononic crystal. Indeed, this silicon-based nanomaterial is crystalline with a power factor that can be optimized by doping using CMOS-compatible processes. Future research on the phononic-crystal electrical conductivity has to be performed in order to compute the full ZT with a good accuracy.