Extracting quantitative oceanographic information from microwave images of the ocean surface requires a physical understanding and an efficient mathematical model of surface wave interaction with currents. In this paper, we consider weak currents (with velocities up to a few tens of centimeters per second) and discuss a perturbation approach that leads to numerically efficient models of surface roughness modulation by the current fields with arbitrary dependence on horizontal coordinates and time. With the wave-atmosphere interaction being described within the relaxation approximation, closed-form analytic expressions are obtained for surface roughness modulation. The hydrodynamic theory is combined with an electromagnetic model based on the small-slope approximation to simulate microwave emission from the ocean surface. Analysis of the theoretical results demonstrates that the physics of surface wave interaction with time-dependent currents, which are inhomogeneous in two spatial dimensions, is more rich and complex than is suggested by the one-dimensional models considered theoretically in the past. Of particular interest for remote sensing is the finding that realistic two-dimensionally inhomogeneous currents, unlike their one-dimensional models, can produce perturbations in microwave brightness temperature that extend well beyond the current field itself. Our model suggests that microwave brightness temperature measurements should be a sensitive tool of observing and quantitatively evaluating surface currents in the ocean.