Abstract Free vibration of two-dimensional functionally graded structures with an exponential material gradation is analyzed in this paper by a meshfree boundary–domain integral equation method. Based on the two-dimensional elasticity theory, boundary–domain integral equations are derived by using elastostatic fundamental solutions. Due to the material inhomogeneity and inertial effect, two domain integrals emerge in the boundary–domain integral equation formulation. Radial integration method is employed to convert the domain integrals into boundary integrals. A meshfree scheme is achieved through approximating the normalized displacements in the domain integrals by a combination of the radial basis functions and the polynomials. Thus, the free vibration problem is reduced into a generalized eigenvalue problem, which involves system matrices with boundary integrals only. By using the developed meshfree boundary–domain integral equation method, free vibration of two-dimensional exponentially graded beams and plates with various material gradients, gradation directions, boundary conditions and aspect ratios is investigated, which demonstrates the high convergence, efficiency and accuracy of the present method.