One-dimensional carbon nanostructures such as nanotubes and nanoribbons can feature near-ballistic electronic transport over micron-scale distances even at room temperature. As a result, these materials provide a uniquely suited solid-state platform for radiation mechanisms that so far have been the exclusive domain of electron beams in vacuum. Here we consider the generation of terahertz light based on two such mechanisms, namely, the emission of cyclotronlike radiation in a sinusoidally corrugated nanowire (where periodic angular motion is produced by the mechanical corrugation rather than an externally applied magnetic field), and the Smith-Purcell effect in a rectilinear nanowire over a dielectric grating. In both cases, the radiation properties of the individual charge carriers are investigated via full-wave electrodynamic simulations, including dephasing effects caused by carrier collisions. The overall light output is then computed with a standard model of charge transport for two particularly suitable types of carbon nanostructures, i.e., zigzag graphene nanoribbons and armchair single-wall nanotubes. Relatively sharp emission peaks at geometrically tunable terahertz frequencies are obtained in each case. The corresponding output powers are experimentally accessible even with individual nanowires, and can be scaled to technologically significant levels using array configurations. These radiation mechanisms therefore represent a promising paradigm for light emission in condensed matter, which may find important applications in nanoelectronics and terahertz photonics.