As Daniel Pomerleano recalled, I have done some work on this with Castano-Bernard. The article is posted here

http://arxiv.org/abs/1301.2930

I can try to summarise the main ideas. Friedman and Tian showed that a set of nodes in X can be smoothed (in the complex category) if the homology classes of exceptional cycles of a resolution satisfy a good relation. Roughly this means that there is a $3$-dimensional chain whose boundary are the exceptional cycles (actually, it's a bit stronger than this...). On the other hand Smith, Thomas and Yau (see reference given by Daniel Pomerleano) have shown that a ``symplectic conifold'' can be resolved (in the symplectic category) if the vanishing cycles (Lagrangian $3$-spheres) satisfy a good relation (i.e. there exists a $4$-chain bounding the spheres).

In the Strominger-Yau-Zaslow philosophy of mirror symmetry one should be able to write $X$ as a torus bundle (with some singular fibres...) $E/L \rightarrow B$. Where $B$ is a real $n$-dimensional manifold, $E$ is a $n$-dimensional vector bundle and $L$ is a maximal lattice ($\cong \mathbb{Z}^n$ over a point). The mirror manifold $X'$ is the dual torus bundle $E'/L' \rightarrow B$.

Now let $n=3$. If one has a one dimensional vector subspace $V_b \subset E_b$, where $b \in B$, then the anhilator of $V_b$ is a $2$-dimensional subvector space $V'_b \subset E'_b$. If you vary $b$ and $V_b$ over some $2$-dimensional object in $B$ then you obtain a $3$-dimensional object in $X$ and on the mirror you have a $4$-dimensional object. This somehow explains the correspondence between "odd cohomology" in $X$ and ``even cohomology'' in $X'$.

In our paper, we construct some $2$-dimensional objects (which we call tropical $2$-cycles) which in $X$ give a $3$-chain whose boundary is the union of the exceptional cycles and in $X'$ give a $4$-chain whose boundary is the union of the vanishing cycles. This proves that if such tropical $2$-cycles exist the obstructions to smoothing $X$ and to resolving $X'$ vanish simultaneously.

Our work relies very much on Mark Gross's ``Topological Mirror Symmetry'', which you can find here

http://arxiv.org/abs/math/9909015