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 (<i>An example of hyperelliptic surfaces with positive index Northeast. Math. J.</i> <b>2</b> (1986), no. 3, 255–257.) found two simply connected complex surfaces $S$ and $S'$ (that is, complex dimension 2), with different Hodge numbers, 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 different 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://iecl.univ-lorraine.fr/~Pierre-Yves.Gaillard/DIVERS/hodgenumbers.pdf