In this paper we investigate the scattering mechanisms of viscous momentum transport in unitary Fermi gases below the pseudogap temperature T* by opening almost all the elastic and inelastic scattering channels. For a given system Hamiltonian, we adopt a microscopic t-matrix approximation to determine the thermodynamical quantities, and we approach an equivalent three-fluid dynamics to calculate the viscous relaxation time τ by two kinds of elementary excitations. The exotic scattering processes raised by uncondensed Fermi pairs greatly decay τ and lead to a universal behavior of τ with temperature, which is closely related to the anomalous transport in a wide range of strongly correlated systems. For linking our findings of τ to the shear viscosity η, we present a Kubo-based expression of η by the stress-tensor correlation function with conserved vertex corrections, and we give an approximate relation for a possible experimental determination of τ. Our results fit well with measurements and are comparable with other theories. Moreover, we verify the newly proposed universal upper bound of the ratio of η and entropy density s in unitary Fermi gases.