This question was [debated in another forum][1] a few years ago.  The result was [a note by Frédéric Campana][2] in which he describes a counterexample as a corollary of another construction.  In 1986 Gang Xiao found two simply connected complex surfaces $S$ and $S'$ (that is, complex dimension 2) that are *homeomorphic* by Freedman's classification.  The homeomorphism has to be orientation-reversing, but $S \times S$ and $S' \times S'$ are orientedly diffeomorphic and of course still have equal Hodge numbers.  Freedman's difficult classification is not essential to the argument, because in 8 real dimensions you can use standard surgery theory to establish the diffeomorphism.

Campana also explains that Borel and Hirzebruch found the first counterexample in 1959, in 5 complex dimensions.

  [1]: http://www.mathkb.com/Uwe/Forum.aspx/research/766/Hodge-numbers
  [2]: http://www.iecn.u-nancy.fr/~gaillard/DIVERS/hodgenumbers.pdf