We analyze Chandra observations of diffuse soft X-ray emission associated with a complete sample of 3CR radio galaxies at z < 0.3. We focus on the properties of the spectroscopic sub-classes of high excitation galaxies (HEGs) and broad line objects (BLOs). Among the 33 HEGs we detect extended (or possibly extended) emission in about 40% of the sources; the fraction is even higher (8/10) when restricting the analysis to the objects with exposure times larger than 10 ks. In the 18 BLOs, extended emission is seen only in two objects; this lower detection rate can be ascribed to the presence of their bright X-ray nuclei that easily outshine any genuine diffuse emission. A very close correspondence between the soft X-ray and optical line morphology emerges. We also find that the ratio between [O III] and extended soft X-ray luminosity is confined within a factor of 2 around a median value of 5. Both results are similar to what is seen in Seyfert galaxies. We discuss different processes that could explain the soft X-ray emission and conclude that the photoionization of extended gas, coincident with the narrow line region, is the favored mechanism. Appendix A is available in electronic form at http://www.aanda.org