We present a new spectral analysis method for the identification of periodic signals in geophysical time series. We evaluate the power spectral density with the adaptive multitaper method, a non-parametric spectral analysis technique suitable for time series characterized by colored power spectral density. Our method provides a maximum likelihood estimation of the power spectral density background according to four different models. It includes the option for the models to be fitted on four smoothed versions of the power spectral density when there is a need to reduce the influence of power enhancements due to periodic signals. We use a statistical criterion to select the best background representation among the different smoothing+model pairs. Then, we define the confidence thresholds to identify the power spectral density enhancements related to the occurrence of periodic fluctuations (γ test). We combine the results with those obtained with the multitaper harmonic F test, an additional complex-valued regression analysis from which it is possible to estimate the amplitude and phase of the signals. We demonstrate the algorithm on Monte Carlo simulations of synthetic time series and a case study of magnetospheric field fluctuations directly driven by periodic density structures in the solar wind. The method is robust and flexible. Our procedure is freely available as a stand-alone IDL code at https://zenodo.org/record/3703168. The modular structure of our methodology allows the introduction of new smoothing methods and models to cover additional types of time series. The flexibility and extensibility of the technique makes it broadly suitable to any discipline.