I also encountered the slow charging problem for several times, regardless of charging with phone turned on or off. Try to turn off the phone, take out the battery, put it back again, lastly charge the battery as usual (I charged with phone turned off). These steps helped my S3 to get back to the normal charging rate. But this is not a solution, the problem will come again. Haiz

IMO, the charging issue is not caused by the cable/hardware. It is more likely caused by the stock firmware, just like the sudden death issue. I believe the charging issue will be fixed too once the sudden death issue is fixed by Samsung.