Hi,
I was staying in the US as a student. My husband and I met in the US in 2005 and we got married in 2007. He was an asylee at that time. I filed for asylee relative petition through my husband and I got approved and we both now have green cards. I did not have anything to do with his political issues and I was not involved in what happened to him in our native country. I still have my valid passport. Can I go and visit my country although my status is asylee through my husband?

Yes, and so can your husband. Confirm this with your immigration attorney. You still need a valid passport, which many refugees and asylees do not have.