Based on lattice dynamics theory and force constant model, we calculated phonon dispersion relation, density of states, and specific heat of graphene ribbons.It is shown that the characteristics of phonon spectra behaves as those of a system between one_dimensional carbon nanotube and two-dimensional graphene. The high and intermediate phonon branches of armchair and zigzag nanoribbons resemble those of zigzag and armchair carbon nanotubes, respectively. With the ribbon width increasing, interestingly, it is found that the low frequency peaks in density of states shifted to lower frequency, indicating a phonon confinement effect. At both low and high temperature,specific heat decreased with the increase of the band width,showing up as a singular quantum size effect.The specific heat at 300K can be fitted by CV=CVg+A/n, where CVg is specific heat of graphene and the A/n term represents the effect of edge on specific heat of ribbons.