  The Rashba spin-orbit coupling in carbon nanotubes and its effect on
spin-dependent transport properties are analyzed theoretically. We focus on
clean non-interacting nanotubes with tunable number of subbands $N$. The
peculiar band structure is shown to allow in principle for Datta-Das
oscillatory behavior in the tunneling magnetoresistance as a function of gate
voltage, despite the presence of multiple bands. We discuss the conditions for
observing Datta-Das oscillations in carbon nanotubes.
