The paper proposes an iterative method for finding the natural frequencies of symmetric oscillations for a rectangular CFCF plate using two hyperbolic-trigonometric series in two coordinates. In this case, a resolving homogeneous infinite system of linear equations which contains the desired natural frequency as a parameter has been obtained. Instead of deriving and solving the frequency equation, it was proposed to use the sequential search for the desired frequency values (the "shooting method") in combination with the method of successive approximations. Numerical results were found for the first three symmetric eigenforms. The influence of the number of series terms and the number of iterations on the accuracy of the results was investigated. The obtained results were compared with those of other authors and the published experimental data.