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For a given complex n-fold M we present an explicit construction of allcomplex (n+1)-folds which are principal holomorphic T2-fibrations over M. Forphysical applications we consider the case of M being a Calabi-Yau 2-fold. Weshow that for such M, there is a subclass of the 3-folds that we construct,which has natural families of non-Kaehler SU(3)-structures satisfying theconditions for N = 1 supersymmetry in the heterotic string theory compactifiedon the 3-folds. We present examples in the aforementioned subclass with M beinga K3-surface and a 4-torus.Comment: LaTeX, 17 pages, organization of the paper was changed, typos  correcte

Geometric Model for Complex Non-K�hler Manifolds with SU (3) Structure