Turaev [defined a simple-homotopy invariant][1] which is a complete invariant of homeomorphism type (originally assuming geometrization). 

Here is the Springer link if you have a subscription:
[Towards the topological classification of geometric 3-manifolds][2]


  [1]: http://www.ams.org/mathscinet-getitem?mr=970081
  [2]: http://www.springerlink.com/content/q3752914430453q7/