The invention provides techniques for securely engaging an organ, such as a beating heart, with a manipulating device. A bag-like device may be deployed around a substantial volume of the organ and the manipulating device. In one embodiment, the bag-like device may be net-like, including one or more mesh apertures. Once deployed, the bag-like device cooperates with the manipulating device when the organ is manipulated and reduces the risk that the organ will be inadvertently released or dropped.