We propose a modified theory of gravitation constructed by the addition of the term f(T mu nu T mu nu) to the Einstein-Hilbert action, and elaborate a particular casef(T mu nu T mu nu) = alpha(T mu nu T mu nu)(eta), where alpha and eta are real constants, dubbed energy-momentum powered gravity EMPG). We search for viable cosmologies arising from EMPG, especially in the context of the late-time accelerated expansion of the Universe. We investigate the ranges of the EMPG parameters (alpha, eta) on theoretical as well as observational grounds leading to the late-time acceleration of the Universe with pressureless matter only, while keeping the successes of standard general relativity at early times. We find that eta = 0 corresponds to the Lambda CDM model, whereas eta not equal 0 leads to a wCDM-type model. However, the underlying physics of the EMPG model is entirely different in the sense that the energy in the EMPG Universe is sourced by pressureless matter only. Moreover, the energy of the pressureless matter is not conserved, namely, in general it does not dilute as rho proportional to alpha(-3) with the expansion of the Universe. Finally, we constrain the parameters of an EMPG-based cosmology with a recent compilation of 28 Hubble parameter measurements, and find that this model describes an evolution of the Universe similar to that in the.CDM model. We briefly discuss that EMPG can be unified with Starobinsky gravity to describe the complete history of the Universe including the inflationary era.