We use high-resolution cosmological hydrodynamical adaptive mesh refinement (AMR) simulations to predict the characteristics of Lyα emission from the cold gas streams that fed galaxies in massive haloes at high redshift. The Lyα luminosity in our simulations is powered by the release of gravitational energy as gas flows from the intergalactic medium into the halo potential wells. The ultraviolet UV background contributes only <20 per cent to the gas heating. The Lyα emissivity is due primarily to electron-impact excitation cooling radiation in gas at ∼2 × 104 K. We calculate the Lyα emissivities assuming collisional ionization equilibrium at all gas temperatures. The simulated streams are self-shielded against the UV background, so photoionization and recombination contribute negligibly to the Lyα line formation. We produce theoretical maps of the Lyα surface brightnesses, assuming that ∼85 per cent of the Lyα photons are directly observable. We do not consider transfer of the Lyα radiation, nor do we include the possible effects of internal sources of photoionization such as star-forming regions. Dust absorption is expected to obscure a small fraction of the luminosity in the streams. We find that typical haloes of mass Mv∼ 1012–1013 M☉ at z∼ 3 emit as Lyα blobs (LABs) with luminosities 1043–1044 erg s−1. Most of the Lyα comes from the extended (50–100 kpc) narrow, partly clumpy, inflowing, cold streams of (1–5) × 104 K that feed the growing galaxies. The predicted LAB morphology is therefore irregular, with dense clumps and elongated extensions. The integrated area contained within surface brightness isophotes of 2 × 10−18 erg s−1 cm−2 arcsec−2 is ∼2–100 arcsec2, consistent with observations. The linewidth is expected to range from 102 to more than 103 km s−1 with a large variance. The typical Lyα surface brightness profile is ∝r−1.2 where r is the distance from the halo centre. Our simulated LABs are similar in luminosity, morphology and extent to the observed LABs, with distinct kinematic features. The predicted Lyα luminosity function is consistent with observations, and the predicted areas and linewidths roughly recover the observed scaling relations. This mechanism for producing LABs appears inevitable in many high-z galaxies, though it may work in parallel with other mechanisms. Some of the LABs may thus be regarded as direct detections of the cold streams that drove galaxy evolution at high z.