The rotational relaxation time, tau, of a spherocylinder and a slightly flexible rod molecule in concentrated solution is computed. The calculations are done by imposing an external flow field that couples to the rotational degrees of freedom. A quasi-equilibrium free energy is obtained in the presence of the flow. Minimization of the free energy in the limit of the zero shear rate yields the concentration dependence of the rotational relaxation time. For spherocylinder tau increases dramatically as the isotropic-nematic phase transition is reached. The values of tau depend not only on the concentration but also on the aspect ratio. It is shown that the effect of flexibility leads to a reduction in persistence length which in turn allows for higher value of the overlap concentration at which the rods start severely hindering each other. For a sufficiently large bending constant kappa the effective persistence length P is found to be P = delta L, where delta approximate to (1-1/2 kappa) and L is the length of the rod.