General aviation instructor pilot reported a near miss with another aircraft during cruise near a non-towered airport. The instructor maneuvered away from the other aircraft to avoid the opposite direction conflict.

Narrative

We were flying southbound from the LGC VOR at 3000 ft. We had been holding at 3000 at the LGC VOR. We made numerous radio calls of what we were doing. The other aircraft also made numerous radio calls that they were inbound for a left downwind for Runway 31 at LGC from 4500 feet. As we departed to the south on a heading of 180 they approached from our right at a higher altitude on the same heading. We had them in sight off our right wing at about 300 feet above and descending. We asked if they had us in sight and got no response. The aircraft then began to turn into us while descending. With a heading aimed directly at us; we took evasive action with full power in a steep climbing left turn to avoid the traffic. In the evasive action we successfully avoided the aircraft. Soon after the aircraft responded that they now had us in sight; but at this point the traffic was no factor with us over 500 feet above and headed in the opposite direction. We were able to avoid a worse situation by taking prompt action when we did not get response from the aircraft leading to a positive outcome.