We present a study of shearfree gravitational collapse using cylindrically symmetric spacetimes whose interior is a non-rotating dissipative fluid bounded by a cylindrical hypersurface beyond which is an Einstein-Rosen vacuum exterior. We consider three different pressure configurations: axially, azimuthally, and radially directed, for which we find new exact interior solutions of the field equations. We show that the matching conditions cannot be satisfied by the fluid with radial pressure, while the axial and azimuthal cases with a lapse function depending only on the time coordinate do satisfy these constraints. We derive, for both cases, a sufficient condition for an emission of gravitational radiation from the interior towards the exterior, We therefore show that, at variance with what happens for spherical symmetry, in the simplified picture of an infinite cylinder of anisotropic shearfree matter, gravitational waves can be emitted during collapsing motion.