We report new deep ALMA observations aimed at investigating the [CII]158um line and continuum emission in three spectroscopically confirmed Lyman Break Galaxies at 6.8<z<7.1, i.e. well within the re-ionization epoch. With Star Formation Rates of SFR ~ 5-15 Msun/yr these systems are much more representative of the high-z galaxy population than other systems targeted in the past by millimeter observations. For the galaxy with the deepest observation we detect [CII] emission at redshift z=7.107, fully consistent with the Lyalpha redshift, but spatially offset by 0.7 (4 kpc) from the optical emission. At the location of the optical emission, tracing both the Lyalpha line and the far-UV continuum, no [CII] emission is detected in any of the three galaxies, with 3sigma upper limits significantly lower than the [CII] emission observed in lower reshift galaxies. These results suggest that molecular clouds in the central parts of primordial galaxies are rapidly disrupted by stellar feedback. As a result, [CII] emission mostly arises from more external accreting/satellite clumps of neutral gas. These findings are in agreement with recent models of galaxy formation. Thermal far-infrared continuum is not detected in any of the three galaxies. However, the upper limits on the infrared-to-UV emission ratio do not exceed those derived in metal- and dust-poor galaxies.