We study the Seiberg-Witten equations on an arbitrary compact complex surface endowed with a Hermitian metric. We obtain a description of the moduli space of solutions in terms of effective divisors on the surface. This result was proved previously in [OT1] in the kähler context. Using concrete examples, we also point out some major differences between the Seiberg-Witten moduli spaces on Kähler resp. non-Kähler surfaces.