  A multinorm one torus associated to a commutative \'etale algebra $L$ over a
global field $k$ is of Kummer type if each factor of $L$ is a cyclic Kummer
extension. In this paper we compute the Tate-Shafarevich group of such tori
based on recent works of Bayer-Fluckiger, T.-Y. Lee and Parimala, and of
T.-Y.~Lee. We also implement an effective algorithm using SAGE which computes
the Tate-Shafarevich groups when each factor of $L$ is contained in a fixed
concrete bicyclic extension of $k$.
