We present results from a continuing interferometric survey of high-redshift submillimeter galaxies (SMGs) with the Submillimeter Array, including high-resolution (beam size ~2 arcsec) imaging of eight additional AzTEC 1.1 mm selected sources in the COSMOS field, for which we obtain six reliable (peak signal-to-noise ratio (S/N) >5 or peak S/N >4 with multiwavelength counterparts within the beam) and two moderate significance (peak S/N >4) detections. When combined with previous detections, this yields an unbiased sample of millimeter-selected SMGs with complete interferometric follow up. With this sample in hand, we (1) empirically confirm the radio-submillimeter association, (2) examine the submillimeter morphology—including the nature of SMGs with multiple radio counterparts and constraints on the physical scale of the far infrared—of the sample, and (3) find additional evidence for a population of extremely luminous, radio-dim SMGs that peaks at higher redshift than previous, radio-selected samples. In particular, the presence of such a population of high-redshift sources has important consequences for models of galaxy formation—which struggle to account for such objects even under liberal assumptions—and dust production models given the limited time since the big bang.