The Johari-Goldstein (JG) secondary relaxation, presumed to be universal, has never been seen in 1,4-polyisoprene (PI) by dielectric spectroscopy, despite very many measurements extending over the past half-century. By using a high-resolution capacitance bridge, we are able to show the existence of a secondary relaxation in PI that has the properties of the JG process. Measurements were also carried out at lower temperatures, which probe the dynamics of chain units "caged" by neighboring segments comprising the local structure. The caged dynamics precede by decades of time the JG relaxation and, from general physical principles, are also expected to be a property of all glass-forming materials. Collectively, the caged dynamics and JG relaxation serve as precursors to structural relaxation (i.e., the glass transition) and thus are of central importance to understanding vitrification. The present data show that the dynamics of caged PI repeat units are manifested as a nearly constant loss (NCL). This NCL has been found in other glass-formers, both molecular and polymeric, and its temperature dependence in PI is similar to that for other materials. The experimental results are consistent with the predictions from the coupling model.