We prove that every family of (not necessarily distinct) even cycles $D_1, \dotsc, D_{\lfloor 1.2(n-1) \rfloor+1}$ on some fixed $n$-vertex set has a rainbow even cycle (that is, a set of edges from distinct $D_i$'s, forming an even cycle). This resolves an open problem of Aharoni, Briggs, Holzman and Jiang. Moreover, the result is best possible for every positive integer $n$.