We present low-dimensional models for the sustenance of exact coherent structures in shear flows of viscoelastic liquids, aimed at helping interpret experiments and direct numerical simulations of turbulent drag reduction by polymers. These models are developed by systematically investigating the effect of incremental amounts of elasticity on the self-sustaining process maintaining the exact coherent structures in shear flows. The recently proposed self-sustaining process for shear flows [F. Waleffe, Phys. Fluids, 9, 83 (1997)] consists of streamwise rolls that lead to redistribution of the mean shear into spanwise streaks. A Kelvin-Helmholtz instability of the spanwise streaky flow then results in the regeneration of the streamwise rolls via nonlinear interactions. Our low-dimensional models enable the identification of the part of the cycle that is interrupted or enhanced by the presence of elasticity. Additionally, we explore the effect of fluid rheology on the flow kinematics, particularly the roles played by the first and second normal stress differences, the uniaxial and biaxial extensional viscosities, as well as shear-thinning.