We theoretically and experimentally explore the emergence of a dynamical density wave (DW) order in a driven dissipative atom-cavity system. A Bose-Einstein condensate is placed inside a high finesse optical resonator and pumped sideways by an optical standing wave. The pump strength is chosen to induce a stationary superradiant checkerboard DW order of the atoms stabilized by a strong intracavity light field. We show theoretically that, when the pump is modulated with sufficient strength at a frequency ω d close to a systemic resonance frequency ω >, a dynamical DW order emerges, which oscillates at the two frequencies ω > and ω < = ω d - ω >. This order is associated with a characteristic momentum spectrum, also found in experiments in addition to remnants of the oscillatory dynamics presumably damped by on-site interaction and heating, not included in the calculations. The oscillating density grating, associated with this order, suppresses pump-induced light scattering into the cavity. Similar mechanisms might be conceivable in light-driven electronic matter.