Kodaira's examples have index $\tau>0$. If $M\to S$ were isotrivial, then it is not hard to see that after pulling back to a finite unramified cover of $S$, the surface becomes a product. But this would force $\tau(M)=0$ [See added note below].
You can look at the book Compact Complex Surfaces by Barth, (Hulek), Peters, and Van de Venn for further explanation.
There are examples of what are sometimes called Kodaira surfaces, where nonisotriviallity is essentially immediate. Namely, find a compact curve $S$ in $M_g$ (which exists once $g>2$), and pull back the "universal" curve.
Added Explanation The index is the signature of the intersection form. By a theorem of Hirzebruch, it can also be computed as
It follows that if $M'\to M$ is a finite unramified cover, then $\tau(M')=0$ if and only if $\tau(M)=0$. In particular, if $M'$ can be chosen as a product of curves, then it can be checked that $\tau(M')=0$, so $\tau(M)=0$.