Using a large sample of spectroscopically confirmed z~ 3 galaxies, we establish an empirical relationship between reddening (E(B- V)), neutral gas covering fraction (f sub(cov)(H I)), and the escape of ionizing (Lyman continuum, LyC) photons. Our sample includes 933 galaxies at z~ 3, 121 of which have deep spectroscopic observations (> ~7 hr) at 850 < ~ lambda sub(rest)< ~ 1300 A with the Low Resolution Imaging Spectrograph on Keck. The high covering fraction of outflowing optically thick H i indicated by the composite spectra of these galaxies implies that photoelectric absorption, rather than dust attenuation, dominates the depletion of LyC photons. By modeling the composite spectra as the combination of an unattenuated stellar spectrum including nebular continuum emission with one that is absorbed by H 1 and reddened by a line-of-sight extinction, we derive an empirical relationship between E(B- V) and f sub(cov)(H 1). Galaxies with redder UV continua have larger covering fractions of H 1 characterized by higher line-of-sight extinctions. We develop a model which connects the ionizing escape fraction with E(B- V), and which may be used to estimate the ionizing escape fraction for an ensemble of galaxies. Alternatively, direct measurements of the escape fraction for our sample allow us to constrain the intrinsic LyC-to-UV flux density ratio to be left angle bracket S(900 A)/S(1500 A)right angle bracket sub(int)> ~ 0.20, a value that favors stellar population models that include weaker stellar winds, a flatter initial mass function, and/or binary evolution. Last, we demonstrate how the framework discussed here may be used to assess the pathways by which ionizing radiation escapes from high-redshift galaxies.