In one dimension, the emergence of non-equilibrium net currents is characteristic for periodically driven systems with broken spatial-reversal symmetry. We show that a similar effect also exists in two-dimensional random systems due to inevitable local symmetry breaking in the potential landscape. We study the properties of the steady-state net currents that emerge in the problem of driven diffusive motion in a random uncorrelated potential under alternating square-wave external field bias. The net current distribution function is found to be approximated well by a Gaussian and the average magnitude of the net currents is defined by higher than linear order even terms in the response function. Furthermore, the net currents can form highly correlated patterns, with large-scale vorticity emerging in the system, in the regime of small and intermediate alternating field amplitudes. With an increase in the amplitude, the vorticity patterns undergo a gradual randomization, approaching a random configuration limit for large values of the external bias. The correlation length decreases as a function of the applied field amplitude and grows nearly linearly with the system size.