Small aircraft Flight Instructor reported an NMAC with an aircraft enroute in cruise flight in the vicinity of SAV VOR.

Narrative

Returning from a training flight from ZZZ1 to ZZZ; a 300NM cross country flight with student performing duties of PIC. An instrument flight plan was filed to ZZZ to ZZZ1 via SAV CRG VORs. Enroute in cruise to SAV; Jacksonville Center advised of VFR traffic 12 o' clock 5500 ft.; type unknown and not in contact with Jacksonville. At this time traffic was level in opposite direction with approximately 3 NM separation to our left. I had advised Jacksonville Center that traffic was in sight and not a factor. A few seconds later the traffic had entered a steep left bank towards us; putting itself on an intersecting course with us and a few seconds after entering the left bank the aircraft began a shallow climb. I took controls from my student when the aircraft banked and when they established and maintained a conflicting course with us; I had initiated a climb. Advised Jacksonville Center of the event and continued the flight as planned.I believe that there are multiple contributing factors that led to this incident: the jet aircraft not utilizing VFR flight following or similar services; and failure to clear their area prior to a maneuver using see and avoid; as well as my designation of the traffic as no factor initially. Also contributing was that my student was using foggles to simulate instrument conditions; due to the need to monitor instruments while my student is flying the aircraft; I was not continuously monitoring the aforementioned traffic; switching my focus inside and outside the airplane. In addition to these factors; the process of evaluating the threat to our flight and safety and weighing that against the need to comply with ATC instruction had delayed my response; climbing the airplane.