We present a study of the rheological and optical behaviour of Kramers bead–rod chains in dilute solution using stochastic computer simulations. We consider two model linear flows, steady shear and uniaxial extensional flow, in which we calculate the non-Newtonian Brownian and viscous stress contribution of the polymers, their birefringence and a stress-optic coefficient. We have developed a computer algorithm to differentiate the Brownian from the viscous stress contributions which also avoids the order (δt)−1/2 noise associated with the Brownian forces. The strain or shear rate is made dimensionless with a molecular relaxation time determined by simulated relaxation of the birefringence and stress after a strong flow is applied. The characteristic long relaxation time obtained from the birefringence and stress are equivalent and shown to scale with N2 where N is the number of beads in the chain. For small shear or extension rates the viscous contribution to the effective viscosity is constant and scales as N. We obtain an analytic expression which explains the scaling and magnitude of this viscous contribution. In uniaxial extensional flow we find an increase in the extensional viscosity with the dimensionless flow strength up to a plateau value. Moreover, the Brownian stress also reaches a plateau and we develop an analytic expression which shows that the Brownian stress in an aligned bead–rod chain scales as N3. Using scaling arguments we show that the Brownian stress dominates in steady uniaxial extensional flow until large Wi, Wi ≈ 0.06N2, where Wi is the chain Weissenberg number. In shear flow the viscosity decays as Wi−1/2 and the first normal stress as Wi−4/3 at moderate Wi. We demonstrate that these scalings can be understood through a quasi-steady balance of shear forces with Brownian forces. For small Wi (in shear and uniaxial extensional flow) and for long times (in stress relaxation) the stress-optic law is found to be valid. We show that the rheology of the bead–rod chain can be qualitatively described by an equivalent FENE dumbbell for small Wi.