A new class of conserved currents, describing nongravitational energy-momentum density, is presented. The proposed currents do not require the existence of a (timelike) Killing vector, and are not restricted to spherically symmetric spacetimes (or similar ones, in which the Kodama vector can be defined). They are based instead on almost-Killing vectors, which could in principle be defined on generic spacetimes. We provide local arguments, based on energy density profiles in highly simplified (stationary, rigidly rotating) star models, which confirm the physical interest of these almost-Killing currents. A mass function is defined in this way for the spherical case, qualitatively different from the Hernández-Misner mass function. An elliptic equation determining the new mass function is derived for the Tolman-Bondi spherically symmetric dust metrics, including a simple solution for the Oppenheimer-Snyder collapse. The equations for the nonsymmetric case are shown to be of a mixed elliptic-hyperbolic nature.