In this paper, the problem addressed is estimation of the false alarm probability, associated with maximum peaks on the periodograms, which are based on the least-squares fitting of the putative periodic signal. This class of periodograms include, for instance, the well-known Lomb-Scargle periodogram and multi-harmonic periodograms. The evaluation of the related distribution of the maximum periodogram values was performed using the so-called Rice method, based on the analysis of the number of up-crossings. This method allowed to obtain simple and closed approximations to the false alarm probability, associated with maximum peaks on the periodograms. The comparison of these analytic approximations with results of Monte Carlo simulations shows that the precision of the analytic expressions should be suitable in the most practical cases.