We present results from a survey for z~2.85 Lyman-continuum (LyC) emission in the HS1549+1933 field and place constraints on the amount of ionizing radiation escaping from star-forming galaxies. Using a custom narrowband filter (NB3420) tuned to wavelengths just below the Lyman limit at z>=2.82, we probe the LyC spectral region of 49 Lyman break galaxies (LBGs) and 91 Ly{alpha} emitters (LAEs) spectroscopically confirmed at z>=2.82. Four LBGs and seven LAEs are detected in NB3420. Using V-band data probing the rest-frame nonionizing UV, we observe that many NB3420-detected galaxies exhibit spatial offsets between their LyC and nonionizing UV emission and are characterized by extremely blue NB3420-V colors, corresponding to low ratios of nonionizing to ionizing radiation (F_UV_/F_LyC_) that are in tension with current stellar population synthesis models. We measure average values of (F_UV_/F_LyC_) for our LBG and LAE samples, correcting for foreground galaxy contamination and H I absorption in the intergalactic medium. We find (F_UV_/F_LyC_)_corr_^LBG^=82+/-45 and (F_UV_/F_LyC_)_corr_^LAE^=7.4+/-3.6. These flux density ratios correspond, respectively, to relative LyC escape fractions of f_esc,rel_^LBG^=5%-8%, and f_esc,rel_^LAE^=18%-49% absolute LyC escape fractions of f_esc,rel_^LBG^=1%-2% and f_esc,rel_^LAE^=5%-15%, and a comoving LyC emissivity from star-forming galaxies of 8.8-15.0x10^24^ erg/s/Hz/Mpc^3^. In order to study the differential properties of galaxies with and without LyC detections, we analyze narrowband Ly{alpha} imaging and rest-frame near-infrared imaging, finding that while LAEs with LyC detections have lower Ly{alpha} equivalent widths on average, there is no substantial difference in the rest-frame near-infrared colors of LBGs or LAEs with and without LyC detections. These preliminary results are consistent with an orientation-dependent model where LyC emission escapes through cleared paths in a patchy interstellar medium.