A valve chip may include a substrate having first and second faces and openings between the first and second faces, and a plurality of flexible valve flaps on one of the faces of the substrate with each flexible valve flap being associated with at least one of the openings. The valve chip may be packaged by forming a frame having an opening therein, and securing the valve chip in the opening of the frame. More particularly, the valve chip may be secured in the opening so that central portions of the first and second faces of the substrate are exposed through the opening in the frame and so that a fluid seal is provided between the frame and edges of the substrate. Related valves, valve assemblies, and methods are also discussed.