Ah, no, actually, in the Christian faith, the body is considered quite significant, because we believe we're going to get it back in the resurrection.
It's just a fairly old belief among Christians that the body should be buried, as opposed to other methods of disposal. It's one of the reasons cremation was actually illegal until recently.
I just want to make it clear that there is no biblical statement on this, so I'm not saying there is anything 'un-Christian' about Christians who do choose to donate organs. It's just a conscience issue.
(And, for those keeping score, I wouldn't accept a donated organ, no. But that's just because I agree, it would be very hypocritical of me not to donate and yet accept an organ for myself)