We study examples of chiral four-dimensional IIB orientifolds with Scherk-Schwarz supersymmetry breaking, based on freely acting orbifolds. We construct a new Z(3) x Z(3)' model, containing only D9-branes, and rederive from a more geometric perspective the known Z(6)' x Z(2)' model, containing D9, D5 and D5 branes. The cancellation of anomalies in these models is then studied locally in the internal space. These are found to cancel through an interesting generalization of the Green-Schwarz mechanism involving twisted Ramond-Ramond axions and 4-forms. The effect of the latter amounts to local counterterms from a low-energy effective field theory point of view. We also point out that the number of spontaneously broken U(1) gauge fields is in general greater than what expected from a four-dimensional analysis of anomalies. (C) 2002 Elsevier Science B.V All rights reserved.