The influence of non-thermal ions on linear dust-acoustic waves is studied in an unmagnetized plasma consisting of ions which have a non-thermal velocity distribution, Boltzmann-distributed electrons and streaming dust particles. A detailed examination is conducted of the dependence of the real frequency and growth rate of the excited instability on the dust drift speed, temperature, particle densities and the parameter $\alpha$ that determines the non-thermal nature of the energetic ions. Comparisons with approximate analytical solutions are also made.