Let $B$ be a variety with torsion canonical sheaf, i.e., $\omega^{\otimes n}_B \cong \mathcal O_B$ for some $n>0$. Then, why does there exist a finite (etale?) morphism $X\to B$ such that $K_X$ is linearly equivalent to zero?

I'm thinking $X\to B$ should be some kind of cyclic covering, but how does one construct such a cyclic covering of $B$ in the first place? And why would it have trivial canonical sheaf?