The derivation of hydrodynamics from the Boltzmann equation in the Anderson-Witting relaxation time approximation assumes the relaxation time to be independent of particle energy, and one is restricted to work in the Landau frame to ensure macroscopic conservation laws. However, the collision time scale typically depends on the microscopic interactions of any real system. We present a framework for the consistent derivation of relativistic dissipative hydrodynamics from the Boltzmann equation with a particle energy-dependent relaxation time by extending the Anderson-Witting relaxation-time approximation and deriving first-order hydrodynamic equations. We show that the obtained transport coefficients have corrections due to the energy dependence of relaxation-time compared to what one gets from the Anderson-Witting approximation of the collision term. We also discuss several interesting scaling features for the ratio of these transport coefficients.