Muslims DO belive in both the Old and New testaments.

They revere Moses, Abraham...etc...and also acknowlege that Jesus was a great prophet, however although they do see these figures as holy and prophets from God, the only word of God to be found untainted, is in the quran...a bit like Christians saying that the Christ has come and the Jews who are still waiting for the messiah.

Being clarifications from the Almighty Allaah, conveyed through His messenger, Muhammad(pbuh), Muslims accept these rectifications to the Christian doctrine without hesitation. If a Christian too were to accept Muhammad (peace be upon him), as a prophet of God, like his belief in Moses or Jesus (peace be upon them), then he should have no difficulty in accepting the above rectification of wrong beliefs and, as a consequence of such acceptance, it will be easy for him to embrace Islaam, thereby placing both Jesus and Mary (peace be upon them) in their true, better and more respectable positions due to them.
http://www.isprin.com/c04.htm