We investigate the observability of cold accretion streams at redshift 3 via Lyα emission and the feasibility of cold accretion as the main driver of Lyα blobs (LABs). We run cosmological zoom simulations focusing on three haloes spanning almost two orders of magnitude in mass, roughly from 1011 to . We use a version of the ramses code that includes radiative transfer of ultraviolet (UV) photons, and we employ a refinement strategy that allows us to resolve accretion streams in their natural environment to an unprecedented level. For the first time in a simulation, we self-consistently model self-shielding in the cold streams from the cosmological UV background, which enables us to predict their temperatures, ionization states and Lyα luminosities with improved accuracy. We find the efficiency of gravitational heating in cold streams in a halo to be around 10–20 per cent throughout most of the halo but reaching much higher values close to the centre. As a result, most of the Lyα luminosity comes from gas which is concentrated at the central 20 per cent of the halo radius, leading to Lyα emission which is not extended. In more massive haloes, of , cold accretion is complex and disrupted, and gravitational heating does not happen as a steady process. Ignoring the factors of Lyα scattering, local UV enhancement and supernovae feedback, the cold ‘messy’ accretion alone in these massive haloes can produce LABs that largely agree with observations in terms of morphology, extent and luminosity. Our simulations slightly and systematically overpredict LAB abundances, perhaps hinting that the interplay of these ignored factors may have a negative net effect on extent and luminosity. We predict that a factor of a few increase in sensitivity from current observational limits should unambiguously reveal continuum-free accretion streams around massive galaxies at z= 3.