Consider　the　scattering　of　a　time-harmonic　elastic　plane　wave　by　a　bi-periodic　rigid　surface.　The　displacement　of　elastic　wave　motion　is　modeled　by　the　three-dimensional　Navier　equation　in　an　unbounded　domain　above　the　surface.　Based　on　the　Dirichlet-to-Neumann　(DtN)　operator,　which　is　given　as　an　infinite　series,　an　exact　transparent　boundary　condition　is　introduced　and　the　scattering　problem　is　formulated　equivalently　into　a　boundary　value　problem　in　a　bounded　domain.　An　a　posteriori　error　estimate　based　adaptive　finite　element　DtN　method　is　proposed　to　solve　the　discrete　variational　problem　where　the　DtN　operator　is　truncated　into　a　finite　number　of　terms.　The　a　posteriori　error　estimate　takes　account　of　the　finite　element　approximation　error　and　the　truncation　error　of　the　DtN　operator　which　is　shown　to　decay　exponentially　with　respect　to　the　truncation　parameter.　Numerical　experiments　are　presented　to　illustrate　the　effectiveness　of　the　proposed　method.