Some aspects of the present invention are directed to unique products and methods for treating fistulae and other passageways and openings in the body. In some preferred forms, an inventive construct will include an elongate graft body that incorporates a plurality of sheet or sheet-like segments. The plurality of sheet or sheet-like segments, in some arrangements, will be stacked in a generally longitudinal direction along the length of the plug body, and optionally, the segments will be received over one or more elongate elements. Elongate elements of this sort can take a variety of forms including a suture, wire, filament, or other relatively thin-bodied elongate member, although in some forms, an elongate element will be or include a somewhat heftier structure such as a biodegradable or non-biodegradable three-dimensional body.