We give conditions on a potentially non-Hausdorff étale groupoid which guarantee that its associated C*-algebra is simple. As a key source of examples, work of Nekrashevych and Exel-Pardo describes a class of C*-algebras arising from the action of a group on a finite alphabet (or more generally, a finite graph). The above authors described these as groupoid C*-algebras and gave conditions which guaranteed their simplicity, usually starting from assumptions which imply the groupoid is Hausdorff. These groupoids need not be Hausdorff, notably for the self-similar action associated to the Grigorchuk group, so it was an open question whether the C*-algebra of the Grigorchuk group action was simple or not. We answer this question in the affirmative. We also discuss simplicity criteria for the associated Steinberg algebras.