  {Abridged} We investigate the observability of cold accretion streams at
redshift 3 via Lyman-alpha (Lya) emission and the feasibility of cold accretion
as the main driver of Lya blobs (LABs). We run cosmological zoom simulations
focusing on 3 halos spanning two orders of magnitude in mass, roughly from
10^11 to 10^13 solar masses. We use a version of the Ramses code that includes
radiative transfer of 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, 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 Lya
luminosities with improved accuracy. We find the efficiency of gravitational
heating in cold streams in a ~10^11 solar mass halo is around 10-20% throughout
most of the halo but reaching much higher values close to the center. As a
result most of the Lya luminosity comes from gas which is concentrated at the
central 20% of the halo radius, leading to Lya emission which is not extended.
In more massive halos, of >10^12 solar masses, cold accretion is complex and
disrupted, and gravitational heating does not happen as a steady process.
Ignoring the factors of Lya scattering, local UV enhancement, and SNe feedback,
cold accretion alone in these massive halos can produce LABs that largely agree
with observations in terms of morphology, extent, and luminosity. Our
simulations slightly and systematically over-predict 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.
