This isn't a proper answer, but here are some remarks.
 
 - Wise's work actually purports to tell you that most (all?) hyperbolic 3-manifold groups *virtually* embed in graph groups - ie, a finite-index subgroup embeds.  So you might lose a lot of torsion when you pass to this finite-index subgroup.
 - [Haglund and Wise][1] have a very nice criterion for embedding non-positively curved cube complexes as subgroups of groups .  It would be completely reasonable to try to use this to construct interesting examples of torsion in subgroups of graph groups.
 - Because the embeddings in Haglund--Wise are *quasiconvex*, it follows from work of Haglund that every such subgroup H is a *virtual retract* - that is, there is a finite-index subgroup K that contains H and the inclusion map has a left inverse K->H.  In particular, any torsion you see in the homology of H will also show up in the homology of K.  It would be easy to use a computer algebra package like GAP to look for torsion in finite-index subgroups of graph groups. 


  [1]: http://www.springerlink.com/content/q4t2224641r6n605/