We propose a linearly implicit structure-preserving numerical method for semilinear Hamiltonian systems with polynomial nonlinearities, combining Kahan's method and exponential integrator. This approach efficiently balances computational cost, accuracy and the preservation of key geometric properties, including symmetry and near-preservation of energy. By requiring only the solution of a single linear system per time step, the proposed method offers significant computational advantages while comparing with the state-of-the-art symmetric energy-preserving exponential integrators. The stability, efficiency and long-term accuracy of the method are demonstrated through numerical experiments on systems such as the Henon-Heiles system, the Fermi-Pasta-Ulam system and the two-dimensional Zakharov-Kuznestov equation.