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The variational nodal method (VNM) is an advanced nodal method in the field of reactor physics. The VNM solves neutron transport equations directly without transverse leakage interpolations used in other nodal methods. Moreover, the VNM can tackle arbitrary expansion orders. Given its benefits, VNM performs well in a variety of applications. However, with the advancement of nuclear science and technology, the ability to handle complex unstructured problems becomes indispensable. In order to increase the geometric compatibility of the VNM, an unstructured-mesh variational nodal method (VNM) for solving multi-dimensional neutron transport equations is presented in this paper. The neutron transport equation is transformed into a variational formulation, in which an even-parity Lagrange multipliers enforce neutron conservation in each node, to obtain the weak-form solution. The entire problem domain is spatially meshed into several unstructured nodes. And the neutron flux, source, and current are spatially approximated by a set of complete orthogonal polynomials. The conception of the unique node is adopted to reduce the memory and improve efficiency. Orthogonal polynomials are established in a standard node for spatial discretization, and each unstructured node is related to the standard node by the coordinate mapping matrix, respectively. The discrete-ordinates (SN) method is employed to deal with angle variables so that the original equation is decoupled into several equations with different discretization directions. Then, the variational formulation is completely discretized, and the response equations are derived. Finally, different types of benchmarks are employed to validate and evaluate the proposed method. Numerical results demonstrate that unstructured-mesh VNM exhibits significant geometric compatibility and comparable accuracy.