Crystalline solids are generally known as excellent heat conductors, amorphous materials or glasses as thermal insulators. It has thus come as a surprise that certain crystal structures defy this paradigm. A prominent example are type-I clathrates and other materials with guest-host structures. They sustain low-energy Einstein-like modes in their phonon spectra, but are also prone to various types of disorder and phonon-electron scattering and thus the mechanism responsible for their ultralow thermal conductivities has remained elusive. Our thermodynamic and transport measurements on various clathrate single crystal series and their comparison with ab initio simulations reveal an all phononic Kondo effect as origin. This insight devises design strategies to further suppress the thermal conductivity of clathrates and other related materials classes, with relevance for thermoelectric waste heat recovery and, more generally, phononic applications. It may also trigger theoretical work on strong correlation effects in phonon systems.