Prosthetic heart valves, which are collapsible to a relatively small circumferential size for less invasive delivery into a patient and which then re-expand to operating size at an implant site in the patient, include a collapsible/expandable stent-like supporting structure and various components of flexible, sheet-like material that are attached to the supporting structure. For example, these sheet-like other components may include prosthetic valve leaflets, layers of buffering material, cuff material, etc. Improved structures and techniques are provided for securing such other components to the stent-like supporting structure of the valve.