So what were these horses? The mythos of the Spanish Jennet is not a modern invention. There are multitudes of records outside of the Iberian peninsula referring to horses an "Spanish Jennets" (or Genets, or Genetas), and they do all share a number of characteristics. Regardless of how they were categorized at home, an Iberian animal whose primary purpose was aristocratic travel was called a "jennet."
The common characteristics were: a "small" head, an abundance of hair, often an unusual coat color, high set neck, and usually an "amble." In paintings they often have long backs, and while we consider this a flaw is does prevent an exuberantly gaiting horse from stepping on itself (ever wonder why you usually see boots on Icelandics?). These were also not exceptionally large animals.
If you're looking for a modern "Jennet," I recommend the Peruvian Paso:
Or a Mangalarga Marchador:
Currently only Pasos (both the Peruvian, and the Puerto Rican Paso Fino) are allowed to be registered with the Spanish Jennet Horse Society which seeks to recreate this historical horse, except in the case of "atigrado," or appaloosa-colored, division where Paso-crosses are permitted.