We present an efficient Galerkin-characteristic finite element method for the numerical solution of convection-diffusion problems in three space dimensions. The modified method of characteristics is used to discretize the convective term in a finite element framework. Different types of finite elements are implemented on three-dimensional unstructured meshes. To allocate the departure points we consider an efficient search-locate algorithm for three-dimensional domains. The crucial step of interpolation in the convection step is carried out using the basis functions of the tetrahedron element where the departure point is located. The resulting semi-discretized system is then solved using an implicit time-stepping scheme. The combined method is unconditionally stable such as no Courant-Friedrichs-Lewy condition is required for the selection of time steps in the simulations. The performance of the proposed Galerkin-characteristic finite element method is verified for the transport of a Gaussian sphere in a three-dimensional rotational flow. We also apply the method for simulation of a transport problem in a three-dimensional pipeline flow. In these test problems, the method demonstrates its ability to accurately capture the three-dimensional transport features.