We present results from the AzTEC/ASTE 1.1-mm imaging survey of 35 Lyα blobs (LABs) found in the SSA22 protocluster at z = 3.1. These 1.1-mm data reach an rms noise level of 0.7–1 mJy beam⁻¹, making this the largest millimetre-wave survey of LABs to date. While one (or possibly two) out of 35 LABs might be detected at 3σ level, no significant (≥3.5σ) emission is found in any of individual 35 LABs. From this, we estimate 3σ upper limits on the far-infrared luminosity of LFIR < 2 × 10¹² L๏ (the dust temperature of 35K and the emissivity index of 1.5 are assumed). Stacking analysis reveals that the 1.1-mm flux density averaged over the LABs is S₁.₁mm < 0.40 mJy (3σ), which places a constraint of LFIR< 4.5 × 10¹¹ L๏. These data constrain the dust spectral energy distributions of the LABs more tightly than ever if their spectral indices at rest-frame wavelength of ≈ 240 μm are similar to those found in (ultra-)luminous infrared galaxies at 0.2 < z < 0.3. Our results suggest that LABs on average have little ultraluminous obscured star formation, in contrast to a long-believed picture that LABs undergo an intense episode of dusty star formation activities with star formation rates of ∼10³M๏ yr⁻¹. Observations with the Atacama Large Millimeter/submillimeter Array are needed to directly study the obscured part of star formation activity in the LABs.