The notion of zitterbewegung is a long-standing prediction of relativistic quantum mechanics. Here we extend earlier theoretical studies on this phenomenon for the case of III-V zinc-blende semiconductors which exhibit particularly strong spin-orbit coupling. This property makes nanostructures made of these materials very favorable systems for possible experimental observations of zitterbewegung. Our investigations include electrons in n-doped quantum wells under the influence of both Rashba and Dresselhaus spin-orbit interactions, and also the two-dimensional hole gas. Moreover, we give a detailed anaysis of electron zitterbewegung in quantum wires which appear to be particularly suited for experimentally observing this effect.