You may want to take a look at the the thesis of Sarah-marie Belcastro (especially p. 135) - toroidalsnark.net/sm_thesis.pdf

@TabesBridges: Let $X$ be the hypersurface and $Y = X \cap (\mathbb P^{r-1} \times \mathbb A^1)$. What is the defining polynomial of $Y$ then? Is it a homogeneous polynomial of degree $r$ in only the variables of $\mathbb P^{r-1}$ or some mixture from those of $\mathbb A^1$? This is a bit confusing to me.

Thanks a lot for your answer - I will study it. In fact, simply-connectedness is not assumed for Calabi-Yau manifolds in the question (only the conditions on cohomologies are assumed). Do you think that there are stll no such fibrations whose general fibers are non-simply connected Calabi-Yau manifolds?

@JasonStarr, your example does not seem to satisfy the second condition.