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.