ARTCC Controller described a loss of separation event alleging the position relief was not completed with everyone involved understanding the traffic and potential conflicts.

Narrative

A B737 and a B767 lost separation. My trainer then gave the B737 a 30 degree left turn. We had just completed a position relief briefing. The RADAR Controller that we relieved talked about the traffic at FL390 but I thought he was talking about another aircraft at FL390. I would say that we probably took the sector earlier than what we should have because the Controller we relieved had several situations occurring that all needing to be taken care of. For example: 1. Traffic at FL360 that needed to get down in a timely matter. 2. A speed overtake on a couple of flights westbound. 3. An aircraft that would usually not be in the stratum at that time; that needed to be descended. I would say because of those situations; along with other factors; contributed to the loss of separation. Recommendation; the Controller that is leaving a sector should take care of the situations before he gives the sector up. Also; the Relieving Controller should not take the sector unless he/she is prepared to take the position.