We prepared hydrogenated butadiene-acrylonitrile (HNBR) elastomer composites with random orientations of carbon nanotubes (CNTs) and aligned CNTs, denoted by random composites and aligned composites, respectively, by means of a simple mechanical blending method. The CNTs were dispersed uniformly in the HNBR matrix in both types of composites. Interestingly, at CNT contents of 1-2.5 vol%, the dielectric loss (tan delta) of the aligned composites increases slightly, and the dielectric constant (epsilon') of aligned composites increases largely with the increasing content of CNTs, whereas both the tan delta and the epsilon' of the random composites increase largely with the increasing content of CNTs. As a result, a high epsilon' (5000 at 1000 Hz) and a low tan delta (0.42 at 1000 Hz) were obtained in the aligned composite with a CNT content of 2.5 vol%, whereas a high epsilon' and a high tan delta were obtained in the random composites. The relationship between the microstructure and dielectric properties was qualitatively analyzed by means of the percolation theory and intercluster polarization model. The mechanism for the achievement of high epsilon' and low tan delta for dielectric composites was discussed. This study provides a guide to design microstructure that yields composites with improved dielectric properties.