  We prove a Fujita-type theorem for varieties with numerically trivial
canonical bundle using properties of semihomogeneous bundles on abelian
varieties. We combine our results with work of Riess on compact hyperk\"{a}hler
manifolds and work of Mukai, Pareschi and Yoshioka to obtain effective global
generation statements for certain moduli spaces of sheaves on abelian surfaces.
Among these is the statment that if $\cL$ is an ample line bundle on the
Hilbert square $S^{[2]}$ of an abelian surface $S,$ then $\cL^{\otimes m}$ is
globally generated for $m \geq 3.$
