Let $X$ and $Y$ be compact complex manifolds. Note that $K_{X\times Y} \cong \pi_1^*K_X\otimes \pi_2^*K_Y$. If $Y$ has trivial canonical bundle, then $K_{X\times Y} \cong \pi_1^*K_X$. Now the pullback of a nef line bundle is again nef, see Proposition 1.8 (i) of *Compact Complex Manifolds with Numerically Effective Tangent Bundles* by Demailly, Peternell, and Schneider. So one can construct many examples by choosing $X$ with $K_X$ nef and $Y$ non-Kähler with $K_Y$ trivial.

**Example:** Let $X$ be a curve of genus $g > 1$ and $Y$ be a primary Kodaira surface. Then $X\times Y$ is a non-Kähler threefold with $K_{X\times Y}$ nef. Note that $X\times Y$ is also not Moishezon as it contains $Y$ as a complex submanifold and $Y$ is not Moishezon.

For more examples of non-Kähler manifolds with $K_Y$ trivial, see the introduction of *Non-Kähler Calabi-Yau Manifolds* by Tosatti. As is pointed out in Proposition 1.1, if $K_Y$ is trivial (or even just torsion), then it admits a metric $h$ with curvature $\Theta_h = 0$ and hence $K_Y$ is nef, so we don't even need to take a product with $X$ in the above construction.