In this article we extend our previous thermodynamic analysis of films confined to slit pores with smooth walls (i.e., plane-parallel solid surfaces without molecular structure) to the situation in which the walls themselves possess structure. Structured-wall models are frequently employed to interpret experiments performed with the surface forces apparatus (SFA), in which thin films (1-10 molecular diameters thick) are subjected to shear stress by moving the walls laterally over one another at constant temperature, chemical potential, and normal stress or load. The periodic structure of the walls is reflected in a periodic variation of the shear stress with the lateral alignment (i.e., shear strain) of the walls. We demonstrate by means of a solvable two-dimensional model that the molecular length scale imposed by the structure of the walls precludes the derivation of a simple mechanical expression for the grand potential analogous to that which herds in the smooth-wall case. This conclusion is borne out by the results of a grand-canonical Monte Carlo simulation of the three-dimensional prototypal model consisting of a Lennard-Jones (12,6) fluid confined between fcc (100) walls. Criteria for the thermodynamic stability of thin films confined by structured walls are derived and applied to the SFA.