Recently developed manufacturing technologies use high pressure and various doping additions to prepare bulk MgB2-based materials with a high critical current density measured by the magnetization method. We use a contactless transformer method, which is based on studying the superconductor response to an induced transport current, to measure AC losses in bulk MgB2 rings synthesized under high pressure. The obtained dependence of the losses on the primary current (applied magnetic field) is fitted by a power law with an exponent of ~2.1 instead of the cubic dependence predicted by Bean’s model and power law electric field–current density (E–J) characteristics with a large exponent. An unusually strong dependence of the AC losses on the frequency is also observed. It is shown that the E–J characteristic of bulk MgB2 is well fitted by the dependence used in the extended critical state model based on account of the viscous vortex motion in the flux flow regime. Numerical simulation using this E–J characteristic gives current and frequency AC loss dependences that agree well with the experimental results.