The transient temperature solution for a functionally graded material (FGM) is formulated by Green’s function based on the Galerkin method. An approximate solution that satisfies the homogeneous boundary condition is substituted into the governing equation to yield an eigenvalue problem. To solve the eigenvalue problem, the eigenfunctions are approximated by a series of polynomials satisfying the homogeneous boundary condition. The Galerkin method is used to determine the coefficients of eigenfunctions. The transient temperature solution for a general heat conduction equation with a source and nonhomogeneous boundary conditions is obtained by using Green’s function, which is expressed by eigenvalues and corresponding eigenfunctions. Transient thermal stresses in a FGM plate and a FGM hollow circular cylinder are discussed.