In this paper we find exact analytic cosmological solutions for a relativistic dissipative fluid, in the framework of the causal Israel–Stewart theory. We use a general expression for the relaxation time, which is related with the bulk viscosity coefficient, the energy density and pressure of the fluid, and non-adiabatic contribution to the speed of sound. Through the factorization method we find some new exact parametric solutions for the special case s= 1 / 2. For each solution the deceleration parameter, the energy density, the dissipative pressure, the entropy, and the ratio between the dissipative pressure and the fluid’s pressure are evaluated as a function of the cosmic time. We finally discuss the kinematic behavior of the solutions and their relationship with their thermodynamic behavior. © 2022, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.