A strategy profile in a repeated game has L bounded recall if play under the profile after two distinct histories that agree in the last L periods is equal.  Mailath and Morris (2002, 2006) proved that any strict equilibrium in bounded-recall strategies of a game with full support public monitoring is robust to all perturbations of the monitoring structure towards private monitoring (the case of  "almost-public monitoring"), while strict equilibria in unbounded-recall strategies are typically not robust. We prove that the perfect-monitoring folk theorem continues to hold when attention is restricted to strategies with bounded recall and the equilibrium is essentially required to be strict.  The general result uses calendar time in an integral way in the construction of the strategy profile.  If the players' action spaces are sufficiently rich, then the strategy profile can be chosen to be independent of calendar time.  Either result can then be used to prove a folk theorem for repeated games with almost-perfect almost-public monitoring.