A PA28 private pilot on a training flight had a close encounter with another Piper low wing aircraft.

Narrative

I was flying on a southbound VFR flight. Upon climb out there was a scattered to broken layer of clouds at 3800 FT AGL; so I elected to remain below the cloud layer and cruise at 3000 FT MSL. I was not receiving traffic advisories because I had not yet requested them. I looked out the left window and saw a white Piper low wing at my altitude on a westerly heading; approximately 500 FT from my position and closing. I reacted with a shallow dive while the second aircraft passed aft of me. From my detection of the traffic to the aircraft passing aft; the incident took less than two seconds. I did not detect any contact between me and the other aircraft; but I elected to divert to a nearby airport for a precautionary landing. I believe the cause of the incident was the absence of radar traffic advisories in a congested training airspace. A contributing factor was the low cloud layer. In the future I believe the flight program should implement an improved plan for maintaining traffic separation among students and other aircraft in the training airspace.