We demonstrate a new technique for femtosecond microfabrication in transparent dielectrics, which employs non-diffracting Bessel-Gauss beams instead of commonly used Gaussian beams. The main advantage achieved this way is the ability to record linear photomodified tracks, extending along the line of non-diffracting beam propagation without sample translation, as would be required for Gaussian beams. The initial near infrared Gaussian laser beam was transformed into the non-diffracting Bessel-Gauss beam by a glass axicon (apex angle 160 deg). The beam was imaged into the bulk of the sample by a telescope consisting oftwo positive lenses, which allowed to vary the focusing cone angle from 5°to 19°, and maximum non-diffracting propagation distance up to 1 cm. We have recorded pre-programmed patterns of multi-shot damage tracks (diameter about 3 µm), extended uniformly along the z-axis by varying the damage spot coordinates in the x - y plane. The experiments were carried out in various transparent dielectrics: silica glass, sapphire, and plexyglass. Physical processes underlying the Gauss-Bessel microfabrication, and its potential applications for stereolithography, 3D microstructuring, and photonic crystal fabrication will be discussed.