Aligned carbon nanotube (A-CNT) based polymer nanocomposites (PNC) are an important class of next-generation advanced materials. Due to the superior, uniaxial properties of the CNTs themselves, composites with tailored, anisotropic properties can be created. However, the experimentally-determined properties of such advanced PNCs, although much higher than the alternatives, still fall short of theoretical predictions. Recently, we have developed the ability to characterize the 3D morphology of such PNCs at the nanoscale using energy-filtered electron tomography. In this presentation, we discuss our novel imaging protocol and an automated segmentation methodology that enables the fast, accurate characterization of statistically-relevant volumes of low-contrast A-CNT PNCs. Further, we discuss image analysis schemes that aid in the extraction of rich, quantitative morphological data (alignment, bundle/network structure, 3D waviness) from the 3D reconstructions. We use this data to re-visit previously-performed experimental measurements of electrical, thermal and mechanical properties in these PNCs. The results show new trends in the dependence of these properties on the CNT volume fraction, and explain previously perplexing phenomena. We observe that the CNT-CNT network connections play a dominant role in transport properties such as electrical and thermal conductivities. Additionally, CNT waviness and proximity are found to strongly influence the PNC stiffness. Electron tomography is shown to be a powerful tool in establishing nanocomposite structure-property relations, making it an essential tool for understanding and tailoring the next generation of advanced materials.