Rapidly quenched quasibinary TiNi–TiCu system alloys with high copper contents (above 20 at.%) exhibit excellent shape memory effect and have considerably narrower hysteresis as compared with the TiNi binary alloy, this advantage being of special importance for cyclic load applications, e.g. for microelectromechanics (MEMS). The aim of this work is to study the effect of annealing parameters and copper content on the shape memory effect in TiNiCu alloys. Thin amorphous ribbons of TiNi-TiCu alloys with copper contents of 25 to 40 at.% were produced by planar flow casting at a melt cooling rate of about 106 K/s. The alloys were crystallized by isothermal annealing with variable duration and by exposing specimens to a short (10 ms) electric pulse. Increasing the copper content to above 30 at.% considerably reduces the plasticity and shape memory effect of the alloys. However, significant reduction of annealing duration greatly improves the shape memory performance due to prevention of the formation of brittle Ti-Cu phases in the alloys structure.