We studied both theoretically and experimentally the influence of the lattice constant on the resonance frequencies of cut-wire-pair structures, which are essential components in assembling left-handed materials. These structures were designed and fabricated, and the transmission spectra were measured in the microwave-frequency regime. Numerical simulation was performed using the transfer-matrix method. All the numerical results were in good agreement with the experimental data. It was found that the lattice parameters of the cut-wire-pair structure provide sensitive effects on the electric resonance, especially, in the propagation k and the electric field E directions, while the magnetic resonance frequency is nearly unchanged, except for its bandwidth and depth. In addition, we also studied the effect of dielectric layer thickness on the resonance frequencies of the cut-wire-pair structure. These sensitive changes in the electric resonance might have remarkable effects on the effective plasma frequency and therefore on the left-handed behavior of combined structures consisting of cut-wire pairs and continuous wires.