Change in damping capacity with strain amplitude was studied in Mg-Al-Si alloy in as-cast, solution-treated and aged states, respectively. The as-cast microstructure of the alloy is characterized by eutectic ${eta}$($Mg_{17}Al_{12}$) phase and Chinese script type $Mg_2Si$ particles. The solution treatment dissolved the ${eta}$ phase into the matrix, while the aging treatment resulted in the distribution of continuous and discontinuous type ${eta}$ precipitates. The solution-treated microstructure showed better damping capacity than as-cast and aged microstructures both in strain-dependent and strain-independent damping regions. The decrease in second-phase particles which weakens the strong pinning points on dislocations and distribution of solute atoms in the matrix, would be responsible for the enhanced damping capacity after solution treatment.