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We derive new exact gravitational wave solutions with dynamical torsion and nonmetricity tensors in the framework of cubic metric-affine gravity (MAG). For this purpose, we consider the full algebraic classification of the gravitational field in general metric-affine geometries and impose a set of type N conditions on the field strength tensors that implement the kinetics of torsion and nonmetricity in a particular cubic MAG model, recently considered to eliminate ghostly instabilities from the vector and axial sectors of the theory. The new solutions represent pp waves characterized by a metric function that includes the dynamical contributions of the torsion and nonmetricity tensors provided by the field equations of the model. In particular, these quantities induce a scalar polarization mode in the gravitational-wave spectrum, thus offering a distinctive phenomenological signature beyond the ordinary tensor polarization modes of general relativity.