In the present work a study is given for the evolution of a flat, isotropic and homogeneous Universe, which is filled with a causal bulk viscous cosmological fluid. We describe the viscous properties by an ultra-relativistic equation of state, and bulk viscosity coefficient obtained from recent lattice QCD calculations. The basic equation for the Hubble parameter is derived by using the energy equation obtained from the assumption of the covariant conservation of the energy-momentum tensor of the matter in the Universe. By assuming a power law dependence of the bulk viscosity coefficient, temperature and relaxation time on the energy density, we derive the evolution equation for the Hubble function. By using the equations of state from recent lattice QCD simulations and heavy-ion collisions we obtain an approximate solution of the field equations. In this treatment for the viscous cosmology, no evidence for singularity is observed. For example, both the Hubble parameter and the scale factor are finite at t = 0, where t is the comoving time. Furthermore, their time evolution essentially differs from the one associated with non-viscous and ideal gas. Also it is noticed that the thermodynamic quantities, like temperature, energy density and bulk pressure remain finite. Particular solutions are also considered in order to prove that the free parameter in this model does qualitatively influence the final results.