A weighting-function model of unsteady skin friction in fully rough-walled flows in one-dimensional ducts is derived using an idealized radial viscosity distribution. The model complements previous work by the authors for smooth-walled flows. It is assumed that, for sufficiently short-lived transients, the viscosity distribution in the cross-section may be regarded as constant and equal to that in a pre-existing steady flow. The eddy viscosity in an outer annulus is assumed to vary linearly from a minimum at the wall to a maximum at the edge of a central core of uniform viscosity. The resulting weighting-function model for short-lived transients is used to develop a simple formula predicting values of unsteady skin friction coefficients suitable for an instantaneous-acceleration model of unsteady skin friction in fully rough pipe flows.