  We establish the existence of rank two Ulrich vector bundles on surfaces that
are either of maximal Albanese dimension or with irregularity 1, under many
embeddings. In particular we get the first known examples of Ulrich vector
bundles on irregular surfaces of general type. Another consequence is that
every surface such that either $q \le 1$ or $q \ge 2$ and its minimal model has
rank one, carries a simple rank two Ulrich vector bundle.
