In this paper, we study the application of a version of the method of simplest equation for obtaining exact traveling wave solutions of the Zakharov-Kuznetsov equation, the modified Zakharov-Kuznetsov equation, and their generalized forms. The Duffing-type equation is used as simplest auxiliary equation. In the meantime, the proposed method is proved to be a powerful mathematical tool for obtaining exact solutions of nonlinear partial differential equations in mathematical physics.