We demonstrate the first three-dimensional simulations of bicontinuous spinodal decomposition of incompressible binary fluid mixtures in a confined geometry. We find that after an initial transient the thickness of a wetting layer L grows with time t as L similar to t(1/3) by a diffusion process and later as L similar to t by a hydrodynamic one. We confirm that the hydrodynamic pumping of fluid via bicontinuous tubes plays dominant roles in the late-stage thickening dynamics of a wetting layer. A domain connected to the wetting layer grows faster than in bulk, which is suggestive of the hydrodynamic origin of the fast-mode kinetics found by Wiltzius and Cumming.