The wide-area imaging surveys with the Herschel Space Observatory at submillimeter (sub-mm) wavelengths have now resulted in catalogs of the order of one-hundred-thousand dusty, starburst galaxies. These galaxies capture an important phase of galaxy formation and evolution, but, unfortunately, the redshift distribution of these galaxies, N(z), is still mostly uncertain due to limitations associated with counterpart identification at optical wavelengths and spectroscopic follow-up. We make a statistical estimate of N(z) using a clustering analysis of sub-mm galaxies detected at each of 250, 350 and 500 μm from the Herschel Multi-tiered Extragalactic Survey centered on the Boötes field. We cross-correlate Herschel galaxies against galaxy samples at optical and near-IR wavelengths from the Sloan Digital Sky Survey, the NOAO Deep Wide Field Survey, and the Spitzer Deep Wide Field Survey. We create optical and near-IR galaxy samples based on their photometric or spectroscopic redshift distributions and test the accuracy of those redshift distributions with similar galaxy samples defined with catalogs from the Cosmological Evolution Survey (COSMOS), which has superior spectroscopic coverage. We model the clustering auto- and cross-correlations of Herschel and optical/IR galaxy samples to estimate N(z) and clustering bias factors. The S 350 > 20 mJy galaxies have a bias factor varying with redshift as b(z) = 1.0+1.0 - 0.5(1 + z)1.2+0.3 - 0.7 . This bias and the redshift dependence is broadly in agreement with galaxies that occupy dark matter halos of mass in the range of 1012 to 1013 M &sun;. We find that galaxy selections in all three Spectral and Photometric Imaging Receiver (SPIRE) bands share a similar average redshift, with langzrang = 1.8 ± 0.2 for 250 μm selected samples, and langzrang = 1.9 ± 0.2 for both 350 and 500 μm samples, while their distributions behave differently. For 250 μm selected galaxies we find the a larger number of sources with z <= 1 when compared with the subsequent two SPIRE bands, with 350 and 500 μm selected SPIRE samples having peaks in N(z) at progressively higher redshifts. We compare our clustering-based N(z) results to sub-mm galaxy model predictions in the literature, and with an estimate of N(z) using a stacking analysis of COSMOS 24 μm detections.