You need to understand that the JDM isn't a real serial port programmer, it doesn't use the serial port at all - it just 'wiggles' the handshake lines, and uses them to create a syncronous port (just as parallel port programmers do).
So using a USB/Serial converter can have very variable results, for a start they run VERY slowly, due to the overheads of USB, and the fact that serial only does a single byte at a time.
Best solution is to get an InchWork ICD2 clone, which is a proper serial port programmer, and will work fine off a USB/Serial converter.