It doesn't have anything to do with the software, but the hardware. There are a few different types of phone connector jacks - stereo, mono, and multi-channel.
Take note of the three bands on the plug and the different colored sockets. This is how you can visually identify the cable has three channels, one for the microphone and two for left and right audio. I suspect this is the kind of connector on your new headphones.
Your audio card is not designed for this kind of multi-channel connection, input and output on the same cable. If the headset didn't come with it, your local electronics store should be able to sell you an appropriate cable; it should have a port for the three-channel line (not that you'll be able to visually inspect it) and then two plugs to connect to the computer's ports.
Just make sure you don't get a simple splitter - it will only connect to the two channels, not all three. Frequently (but not always) the plugs will be different colors if it's what you need, but the same if it's just a splitter.