We investigate the redshift dependence of the contribution to the extragalactic far-infrared/submillimeter background from galaxies detected by the Spitzer Space Telescope at 8 and 24 μm. Using seven-band optical to mid-infrared photometry, we estimate photometric redshifts for the Spitzer sources that appear to be mostly L* galaxies at a median redshift of z = 1.0. These sources, extracted from deep 8 and 24 μm mosaics of the CUDSS 14 hour field with 5 σ limits of 5.8 and 70 μJy, respectively, exhibit significant 850 and 450 μm emission as observed by SCUBA. At 850 μm, after removing ≥4 σ sources and those securely identified in our companion cross-matching paper, we measure stacked flux at the significance level of 4.4 and 2.9 σ from the full 8 and 24 μm galaxy catalog, respectively. At 450 μm, flux is detected from all 8 μm galaxies at the level of 3.5 σ, while there is no significant emission from the 24 μm galaxies. We find that the 850 μm flux is emitted almost exclusively at z 1.3 from the Spitzer sources with 0.44 mJy (4.7 σ) per 8 μm source, and 0.51 mJy (2.8 σ) per 24 μm source. This corresponds to a contribution of (16 ± 3)% toward the 850 μm extragalactic background from the 8 μm sources, and (5.0 ± 1.8)% from the 24 μm sources. At 450 μm, only the 8 μm sources within the redshift interval 1 < z < 2 exhibit significant emission with an average flux per source of 3.35 mJy (3.0 σ). This is a contribution of (37 ± 12)% to the 450 μm background.