The transient stress response of homeotropic monodomains of dilute nematic solutions of side-chain liquid-crystalline polysiloxanes (LCPs) in 4,4＇-(n-pentyloxy)cyanobiphenyl (5OCB) is investigated. In contrast to the flow-aligning behavior of 5OCB, strong stress oscillations are observed for the dilute LCP solutions, characteristic of flow-tumbling behavior. The dependence of the oscillation periodicity on LCP concentration and LCP structure (molecular weight and spacer length) is presented. From analysis of the stress transients, the Leslie viscosity coefficients, alpha(2) and alpha(3), are calculated. For all LCP structures, the increment delta alpha(2) on dissolution of LCP is negative, whereas the increment delta alpha(3) is positive. The magnitude and signs of these viscosity increments are strictly inconsistent with earlier electrorheological measurements of the Miesowicz viscosity increments, delta eta(b) and delta eta(c), of these solutions, using as a guide a hydrodynamic model that computes the contribution of the chains to viscous dissipation in a simple shear flow. We propose that an additional dissipation mechanism should be considered, which derives from the presence of an elastic torque between director rotation and LCP orientation. Corresponding modification of the theory leads to improved agreement with experiment.