This paper considers dynamical properties of oscillatory phenomena in gene regulatory networks with cyclic feedback. Specifically, we provide analytic estimates of oscillation profiles such as frequency, phase and amplitude based on the harmonic balance method, which is one of the frequency domain techniques to examine nonlinear oscillatory behaviors by approximating with bias and first order harmonic components. These analytic estimates allow us to find essential physical quantities that primarily determine the oscillation profiles, and they provide a guidance for synthetic oscillator design. We demonstrate the effectiveness of the proposed approach through illustrative numerical simulations.