We consider polling systems in which a single server visits stations in cyclic order and serves customers at each station according to either the gated rule or the exhaustive rule of the station. There are multiple classes of customers at each station, and they are served in either the priority order or the first-come-first-served (FCFS) order. After completing a service at a station, each customer may be routed to one of the stations or leave the system according to the Markovian feedback mechanism. In this paper, a new approach to mean sojourn times in multiclass queues, developed in [11, 14, 15], is extended to the feedback polling systems as follows. We define the conditional expected sojourn times and find their linear functional expressions by solving some equations. The steady state average sojourn times are derived from these expressions by simple limiting procedures, and their values are obtained by solving a set of linear equations. We also consider composite scheduling algorithms and calculate mean path times.