Suppose $M$ is a Moishezon manifold with $c_1(M)=0$ in $H^2(M,\mathbb{R})$. Does it follow that $K_M$ is torsion in $\mathrm{Pic}(M)$?

This is true whenever $M$ is Kähler (and therefore projective) and was proved independently by Bogomolov, Fujiki and Lieberman. It is also a well-known consequence of Yau's solution of the Calabi Conjecture. Also when $\mathrm{dim}M=2$ then $M$ is automatically projective, so the question is really about dimensions $3$ or more.

The only examples that I know of non-Kähler Moishezon manifolds with $c_1(M)=0$ are obtained by applying a Mukai flop to a projective hyperkähler manifold, and so they have holomorphically trivial canonical bundle. They are described here.

Are there other simple examples of such manifolds?

The same question can also be asked for compact complex manifolds bimeromorphic to Kähler (i.e. in Fujiki's class $\mathcal{C}$).