We study the zero-temperature stochastic Ising model on some connected planar quasi-transitive graphs, which are invariant under rotation and translation. The initial spin configuration is distributed according to a Bernoulli product measure with parameter $ p\in(0,1) $. In particular, we prove that if $ p=1/2 $ and the graph underlying the model satisfies the planar shrink property (which causes each finite cluster to shrink to a site and then vanish with positive probability) then all vertices flip infinitely often almost surely.