Well, if there is only one 3.5mm port on your computer it may not be a combo, input/output, port. Many computers have an output only one. Also, in some cases, the input/output combo port will not work with a splitter. Happens a lot.
One way to see if this is the problem is to get a 3.5mm to USB adapter and try using the microphone through it. They are less touchy and less problematic. They are pretty cheap (the adapters) or you can get a microphone or headset/microphone combo that comes with a USB.