I can't tell wether the "Taiwan-Cable" I have used was a "DLR-3P" type or something else.

But I think it does not really matter as you are saying your cable works with "HyperTerm".

Do you have a DB9 "Breakout-Box" handy?

If so:
connect your DLR-3P cable to your PC across "Breakout-Box"
and find out which pins are really needed by checking communication with "Hyperterm"

Once you have identified the Pins check the Voltage Level at the "Supply" pins (DTR and/or RTS) it is most likely something between +10 an +12 V.

Connect RX, TX and GND to your PIC
and supply the identified "supply"-pins with the correct voltage.

Assuming your PIC-code is ok, It should work.

If it doesn't work, try to communicate with your PIC using "Hyperterm"
and debug the PIC code.

Questions:

What type of PIC are ypu using?

Are you using the HW USART and HSERIN/HSEROUT?

Do you have a "MAX232" type of level shifter connected to the PIC?

What is the PICs clock speed?
(9600 Baud won't work with a 4MHz clock.)

regards

Ralph