  We study SYZ mirror symmetry in the context of non-Kaehler Calabi-Yau
manifolds. In particular, we study the six-dimensional Type II supersymmetric
$SU(3)$ systems with Ramond-Ramond fluxes, and generalize them to higher
dimensions. We show that Fourier-Mukai transform provides the mirror map
between these Type IIA and Type IIB supersymmetric systems in the semi-flat
setting. This is concretely exhibited by nilmanifolds.
