This is probably not the most optimal way to do it, but this was what came to mind right away. Also, this is in some sense more general than what you ask and in some sense it is less. Finally, it is not a complete solution, but it might give you some ideas.

So, let $M$ and $N$ be complex projective manifolds.

According to Shokurov's rational connectedness conjecture the set $f^{-1}(x)$ is rationally chain connected. This was proved by Hacon and McKernan [here][1]. Rationally connected manifolds are simply connected. This is due to Campana and Kollár-Miyaoka-Mori and can be found for example in [this][2] book. 

You are not there yet, but I think you should be able to prove that the graph of the components of $f^{-1}(x)$ should be a tree and that the individual components are simply connected using the above argument. Check Kollár's article in the [book][2] as well.


  [1]: http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.dmj/1178738561&page=record
  [2]: http://books.google.com/books?id=ruDDjLb4d9gC&pg=PA124&lpg=PA124&dq=rationally+connected+varieties+are+simply+connected&source=bl&ots=jVn6lEpWRi&sig=4JzAr2D8MYUOC9xpF5QL4_yWYm8&hl=en&ei=FtkcTafxLY64sQO7_di4Cg&sa=X&oi=book_result&ct=result&resnum=3&ved=0CCQQ6AEwAg#v=onepage&q=rationally%2520connected%2520varieties%2520are%2520simply%2520connected&f=false