We consider directed quantum transport in one-dimensional, periodic tight-binding lattices that are driven by unbiased., but asymmetric driving forces which are composed of a fundamental and its second harmonic (harmonic mixing signal). ne reduction to a single-band dynamics then yields a formally exactly integrable dynamics. An asymmetric drive generally breaks time-reversal symmetry without causing any dissipation. For an initially localized particle with zero momentum no finite coherent current emerges; in contrast, a directed current occurs. however, when prepared with an initial coherence. The presence of phase fluctuations yields a decaying current, thereby rendering the nondissipative currents a transient phenomenon. In contrast, the presence of quantum dissipation yields at all temperatures a finite asymptotic current independent of preparation.