In a Debye-Yukawa system, charged micro-particles interact with each other through a screened Coulomb potential phi = (Q/r)exp{-r /lambda (D)} where Q is the charge and lambda (D) is the screening length. In a strongly coupled Debye-Yukawa system (where Q is much larger than the magnitude of an electron charge), a plasma crystal may be formed. One important feature of such a strongly coupled system is the existence of transverse waves (or shear waves) in low frequency modes. In this paper we present an analytic treatment of longitudinal (or compressional) and transverse low frequency modes in two-dimensional plasma crystals. Dispersion relations of the modes are obtained for two-dimensional triangular lattices. It is also found that the modes (the compressional and the transverse) are strongly damped near the 'negative dispersion' regime. Theoretical predictions agree quantitatively with recent experiments, and the effects of neutral gas damping are also discussed.