Different types of waveguide leaky-wave antennas, asymmetrically perturbed with printed-circuits, are studied in this paper. The capability to modify the leakage constant of the excited leaky-wave mode, while maintaining unchanged its phase constant, is studied in detail for each design. Several slot and strip configurations are proposed, in which the width and the position of the planar printed circuit perturbation is modified along the length of the antenna to obtain different tapering topologies. All of them are mechanically flexible to realize, thus simplifying the manufacturing process for millimeter wavelengths applications. The two-dimensional method used for the analysis is accurate for any geometry of the planar perturbation, and also allows for the computation of the whole spectrum of modes in these open waveguides. This makes possible to predict the appearance of unwanted channel-guide modes, and also allows for the study of the coupling effect between the desired leaky-wave mode and them. A tapered design is performed and compared with full-wave three-dimension simulations to validate the proposed technology.