Many one-dimensional systems of experimental interest possess multiple bands arising from shallow confining potentials. In this paper, we study a gas of weakly interacting fermions and show that the bulk viscosity is dramatically altered by the occupation of more than one band. The reasons for this are twofold: a multichannel system is more easily displaced from equilibrium and the associated relaxation processes lead to more rapid equilibration than in the single channel case. We estimate the bulk viscosity in terms of the underlying microscopic interactions. The experimental relevance of this physics is discussed in the context of quantum wires and trapped cold atomic gases.