The anisotropic behaviour of the matter diffusion in columnar discotic liquid crystals is studied by the 'mirage effect' technique. The Dh and Do mesophases of C&ET and C1 IHET, respectively, are considered. The impurity (a dye, the l-[4-(xylylazo)xylylazo]-2-naphthol) diffusing in these mesophases is detected by the photothermal deflection technique. Measurements of the diffusion coefficients are performed in two perpendicular directions, along and perpendicularly to the molecular columns. Effects of impurity size, length and type of the branched chains on the discs of triphenylene, and molecule stackings in columns, are presented.