This paper is concerned with the stability and accuracy of finite-element analyses of transient thermoviscoelasticity subjected to irreversible thermodynamic processes. The study includes the use of an incremental free energy that incorporates internal state variables in terms of a discretized generalized Maxwell model. Numerical integration is then performed, with past histories being updated in each discretized time step. The effects of thermomechanical coupling, internal dissipation, boundary conditions, and various temporal operators are studied with respect to spectral radii, phase errors, numerical damping, etc. A stability criterion predicted by spectral radii for combined equations of motion and heat conduction confirms the consequent numerical results for transient response according to the estimated degree of stability. Stable solutions generally lead to convergence to the exact solution, but with lower rates of convergence than those of linear-uncoupled equations. For simplicity, the numerical examples deal with one-dimensional structures.