We characterise simplicity of twisted C∗-algebras of row-finite κ-graphs with no sources. We show that each 2-cocycle on a cofinal κ-graph determines a canonical secondcohomology class for the periodicity group of the graph. The groupoid of the κ-graph then acts on the cartesian product of the infinite-path space of the graph with the dual group of the centre of any bicharacter representing this second-cohomology class. The twisted κ-graph algebra is simple if and only if this action is minimal. We apply this result to characterise simplicity for many twisted crossed products of κ-graph algebras by quasifree actions of free abelian groups.