We introduce and study the behavior of a tethered membrane of non-zero thickness embedded in three dimensions subject to an effective self-attraction induced by hydrophobicity arising from the tendency to minimize the area exposed to a solvent. The phase behavior and the nature of the folded conformations are found to be quite distinct in the small and large solvent size regimes. We demonstrate spontaneous symmetry-breaking with the membrane folding along a preferential axis, when the solvent molecules are small compared to the membrane thickness. For large solvent molecule size, a local crinkling mechanism effectively shields the membrane from the solvent, even in relatively flat conformations. We discuss the binding/unbinding transition of a membrane to a wall that serves to shield the membrane from the solvent. ; Comment: 7 pages, 5 figures, to appear in EPL