In the present paper, we introduce the H2-wavelet method for the fast solution of nonlocal operator equations on unstructured meshes. On the given mesh, we construct a wavelet basis which provides vanishing moments with respect to the traces of polynomials in the space. With this basis at hand, the system matrix in wavelet coordinates is compressed to O(N log N) relevant matrix coefficients, where N denotes the number of boundary elements. The compressed system matrix is computed with nearly linear complexity by using the H2-matrix approach. Numerical results in three spatial dimensions validate that we succeeded in developing a fast wavelet Galerkin scheme on unstructured triangular or quadrangular meshes.