Novel seasonal hidden Markov models (SHMMs) for stochastic time series with periodically varying characteristics are developed. Nonlinear interactions among SHMM parameters prevent the use of the forward-backward algorithms which are usually used to fit hidden Markov models to a data sequence. Instead, Powell's direction set method for optimizing a function is repeatedly applied to adjust SHMM parameters to fit a data sequence. SHMMs are applied to a set of meteorological data consisting of 9 years of daily rain gauge readings from four sites. The fitted models capture both the annual patterns and the short term persistence of rainfall patterns across the four sites.