This article describes the first steps toward comprehensive characterization of molecular transport within scaffolds for tissue engineering. The scaffolds were fabricated using a novel melt electrospinning technique capable of constructing 3D lattices of layered polymer fibers with well-defined internal microarchitectures. The general morphology and structure order was then determined using T2-weighted magnetic resonance imaging and X-ray microcomputed tomography. Diffusion tensor microimaging was used to measure the time-dependent diffusivity and diffusion anisotropy within the scaffolds. The measured diffusion tensors were anisotropic and consistent with the cross-hatched geometry of the scaffolds: diffusion was least restricted in the direction perpendicular to the fiber layers. The results demonstrate that the cross-hatched scaffold structure preferentially promotes molecular transport vertically through the layers (z-axis), with more restricted diffusion in the directions of the fiber layers (x-y plane). Diffusivity in the x-y plane was observed to be invariant to the fiber thickness. The characteristic pore size of the fiber scaffolds can be probed by sampling the diffusion tensor at multiple diffusion times. Prospective application of diffusion tensor imaging for the real-time monitoring of tissue maturation and nutrient transport pathways within tissue engineering scaffolds is discussed.