We compare high-resolution optical and radio imaging of 12 luminous submillimeter galaxies at a median z = 2.2 ± 0.2 observed with the Hubble Space Telescope (HST) and the MERLIN and Very Large Array (VLA) radio interferometers at comparable spatial resolution, ~03 (~2 kpc). The radio emission is used as a tracer of the likely far-IR morphology of these dusty, luminous galaxies. In ~30% of the sample the radio emission appears unresolved at this spatial scale, suggesting that the power source is compact and may be either an obscured active galactic nucleus or a compact nuclear starburst. However, in the majority of the galaxies (8/12; ~70%), we find that the radio emission is resolved by MERLIN/VLA on scales of ~1'' (~10 kpc). For these galaxies we also find that the radio morphologies are often broadly similar to their rest-frame-UV emission traced by our HST imaging. To assess whether the radio emission may be extended on even larger scales (1'') resolved out by the MERLIN+VLA synthesized images, we compare VLA B-array (5'' beam) to VLA A-array (15 beam) fluxes for a sample of 50 μJy radio sources, including five submillimeter galaxies. The submillimeter galaxies have comparable fluxes at these resolutions, and we conclude that the typical radio-emitting region in these galaxies is unlikely to be much larger than ~1'' (~10 kpc). We discuss the probable mechanisms for the extended emission in these galaxies and conclude that their luminous radio and submillimeter emission arises from a large, spatially extended starburst. The median star formation rates for these galaxies are ~1700 M yr^-1 (M > 0.1 M), occurring within regions with typical sizes of ~40 kpc² and giving a star formation density of 45 M yr^-1 kpc^-2. Such vigorous and extended starbursts appear to be uniquely associated with the submillimeter population. A more detailed comparison of the distribution of UV and radio emission in these systems shows that the broad similarities on large scales are not carried through to smaller scales, where there is rarely a one-to-one correspondence between the structures seen in the two wave bands. We interpret these differences as resulting from highly structured internal obscuration within the submillimeter galaxies, suggesting that their vigorous activity is producing windblown channels through their obscuring dust clouds. If correct, this underlines the difficulty of using UV morphologies to understand structural properties of this population and also may explain the surprising frequency of Lyα emission in the spectra of these very dusty galaxies.