Air-vented liquid dispensers are disclosed herein, including refill units for use in connection with such dispensers. A rigid container holds a liquid and has a neck portion. A manifold is secured to the neck portion underneath the container, and includes an air passage for air to enter into the container and a liquid passage for air to exit the container. A gasket member is located between the container neck and the manifold, and has a flexible and resilient flap portion which acts as a check valve for the air passage. An air tube carries the air up into the rigid container.