Context. The reionization of the Universe is one of the most important topics of present-day astrophysical research. The most plausible candidates for the reionization process are star-forming galaxies, which according to the predictions of the majority of the theoretical and semi-analytical models should dominate the H I ionizing background at z greater than or similar to 3. Aims. We measure the Lyman continuum escape fraction, which is one of the key parameters used to compute the contribution of star-forming galaxies to the UV background. It provides the ratio between the photons produced at lambda <= 912 angstrom rest-frame and those that are able to reach the inter-galactic medium, i.e. that are not absorbed by the neutral hydrogen or by the dust of the galaxy's inter-stellar medium. Methods. We used ultra-deep U-band imaging (U = 30.2 mag at 1 sigma) from Large Binocular Camera at the Large Binocular Telescope (LBC/LBT) in the CANDELS/GOODS-North field and deep imaging in the COSMOS and EGS fields in order to estimate the Lyman continuum escape fraction of 69 star-forming galaxies with secure spectroscopic redshifts at 3.27 <= z <= 3.40 to faint magnitude limits (L = 0.2L*, or equivalently M-1500 similar to -19). The narrow redshift range implies that the LBC U-band filter exclusively samples the lambda <= 912 angstrom rest-frame wavelengths. Results. We measured through stacks a stringent upper limit (<1.7% at 1 sigma) for the relative escape fraction of H I ionizing photons from bright galaxies (L > L*), while for the faint population (L = 0.2L*) the limit to the escape fraction is less than or similar to 10%. We computed the contribution of star-forming galaxies to the observed UV background at z similar to 3 and find that it is not sufficient to keep the Universe ionized at these redshifts unless their escape fraction increases significantly (>= 10%) at low luminosities (M-1500 >= -19). Conclusions. We compare our results on the Lyman continuum escape fraction of high-z galaxies with recent estimates in the literature, and discuss future prospects to shed light on the end of the Dark Ages. In the future, strong gravitational lensing will be fundamental in order to measure the Lyman continuum escape fraction down to faint magnitudes (M-1500 similar to -16) that are inaccessible with the present instrumentation on blank fields. These results will be important in order to quantify the role of faint galaxies to the reionization budget.