The motion of singular quantized vortex lines in superfluid 3He is considered for the A and B phases. Mutual friction is calculated within a microscopic quantum-mechanical Green's-function formalism, valid for dynamical processes. This enables us to include all the different physical phenomena in a unified approach. We consider axisymmetric vortices for temperatures considerably lower than Tc. In this regime, the main contribution to the force exerted on a moving vortex originates from the localized Fermi excitations occupying quantized energy eigenstates in the vortex core. These 3He quasiparticle states are similar to the quantized motion of charge in a magnetic field; thus vortex motion in 3He resembles the Hall phenomenon in metals. The outcome is that the viscous drag cannot simply be expressed through the cross sections for 3He quasiparticles scattering off the vortex, but is rather due to the mutual interactions between the localized quasiparticles and the normal excitations. Our calculations conform with the experimental values for the mutual-friction parameters. We also discuss vortex oscillations, and predict that strong dissipation should be observed at a resonant frequency of about 10 kHz, owing to transitions between the bound-state energy levels. This effect could be used for detecting and measuring the quantization of the bound-state spectrum for superfluid 3He in the vortex-core matter.