Recent experiments have shown that first-order spin transitions can occur in polymeric FeII compounds. In such materials the metal ions are linked by ligand groups to form polymeric chains. Existing theories of the spin transition in crystalline materials are discussed, but we argue that the physical justification of these theories may be inappropriate for polymeric compounds. We describe a possible mechanism in which local strain in the ligand network can drive the transition. A model Hamiltonian is presented and solved, and the effects of pressure and dopant concentration on the spin transition are investigated. We also point out that the model describes many of the effects observed in experiments on crystalline materials.