We consider the multilinear polytope which arises naturally in binary polynomial optimization. Del Pia and Di Gregorio introduced the class of odd $β$-cycle inequalities valid for this polytope, showed that these generally have Chv{á}tal rank 2 with respect to the standard relaxation and that, together with flower inequalities, they yield a perfect formulation for cycle hypergraph instances. Moreover, they describe a separation algorithm in case the instance is a cycle hypergraph. We introduce a weaker version, called simple odd $β$-cycle inequalities, for which we establish a strongly polynomial-time separation algorithm for arbitrary instances. These inequalities still have Chv{á}tal rank 2 in general and still suffice to describe the multilinear polytope for cycle hypergraphs. Finally, we report about computational results of our prototype implementation. The simple odd $β$-cycle inequalities sometimes help to close more of the integrality gap in the experiments; however, the preliminary implementation has substantial computational cost, suggesting room for improvement in the separation algorithm.