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Recently, the interest in obtaining materials with a large magnetodielectric effect has greatly increased due to their potential applications in novel devices such as tunable filters, four-state memories, magnetic sensors, and spin-charge transducers. We have experimentally studied the magnetodielectric properties of magnetorheological elastomers in stepwise time-varying DC magnetic fields at room temperature. These composite materials contain micrometer-sized iron particles dispersed in compliant elastomer matrices. The ferromagnetic filler particles can rearrange in external DC magnetic fields. This leads to significant modulation of their physical properties. It is found that the imposition of a magnetic field significantly increases both the effective dielectric constant of these composite materials as well as their effective conductivity. These magnetodielectric effects are more pronounced for larger concentrations of soft-magnetic filler particles and softer elastomer matrices. The largest observed relative change of the effective dielectric constant in the maximum magnetic field of about 0.6 T is of the order of 1000%. In particular, we measured the largest, to the best of our knowledge, magnetodielectric effect in polymer-based composite materials at room temperature. The largest observed absolute change of the loss tangent is approximately 0.8. The transient response of the magnetodielectric effect to step magnetic-field excitations is studied in dependence on the step amplitude. A significant hysteresis of the magnetodielectric effect on the externally applied magnetic field is observed. The findings are attributed to the rearrangement of ferromagnetic filler particles in external magnetic fields.