Thermal conductivity in dielectric crystals is the result of the relaxation of lattice vibrations described by the phonon Boltzmann transport equation. Remarkably, an exact microscopic definition of the heat carriers and their relaxation times is still missing: Phonons, typically regarded as the relevant excitations for thermal transport, cannot be identified as the heat carriers when most scattering events conserve momentum and do not dissipate heat flux. This is the case for two-dimensional or layered materials at room temperature, or three-dimensional crystals at cryogenic temperatures. In this work, we show that the eigenvectors of the scattering matrix in the Boltzmann equation define collective phonon excitations, which are termed here "relaxons". These excitations have well-defined relaxation times, directly related to heat-flux dissipation, and they provide an exact description of thermal transport as a kinetic theory of the relaxon gas. We show why Matthiessen's rule is violated, and we construct a procedure for obtaining the mean free paths and relaxation times of the relaxons. These considerations are general and would also apply to other semiclassical transport models, such as the electronic Boltzmann equation. For heat transport, they remain relevant even in conventional crystals like silicon, but they are of the utmost importance in the case of two-dimensional materials, where they can revise, by several orders of magnitude, the relevant time and length scales for thermal transport in the hydrodynamic regime.