seems like you have a headset with a 4pole 3.5mm connector. is this correct?
http
/acc3658.imghost.cafe24.com/acc/home-e/a_qna.jpg 3pole vs 4pole (ignore the diameter part)
also, you are talking about your new pc having a microphone jack AND a seperate headphone jack. is this correct?
if yes...if you have a 4pole like i think you do and two connections like i said above then you can just get a phone headset to pc adapter (or the other various names its known as such as 3.5mm 4pole to two 3pole, console headset to pc adatper, basically the cables are the same thing just with different words..)