A ventable container for materials to be heated (preferably digested) by microwave radiation includes a movable rupture diaphragm in a venting passageway in a closure or cover for such container. Such diaphragm is tightenable into place so that the container is sealed and in such sealed state is protected by the rupture diaphragm against pressures that could develop in the container that are higher than that for which it was designed. An important feature of the invention is that the rupture diaphragm is controllably movable, when desired, to open positions to vent the container during heating and afterward, and is resealable after such venting. Because of the chemical reactivity of the reagents utilized in digestions an inner chemically resistant liner is desirably employed in conjunction with a physically stronger outer body, with the outer body giving the liner enough strength to withstand pressures developed during the heating. Also within the invention are processes for heating and digesting materials in the described containers with microwave radiation, microwave heating apparatuses including the invented containers, and lined containers wherein the liners are of improved sealing structures so that the containers are more tightly sealed when heated.