And of course, I couldn't stop to consult at Wikipedia
Certain groups, like the Roma (gypsies), oppose organ donation on religious grounds, but most of the world's religions support donation as a charitable act of great benefit to the community.
(...) Jewish medical ethics takes a unique approach. It accepts organ donation as a meritorious charitable act, but with two conditions: that the donor be deceased before removal of the organ and that the organ be treated respectfully (and not, for instance, merely discarded if it for some reason becomes unusable). The ethical problem stems from a lack of consensus on the definition of "deceased." According to the strictest interpretation of halachah, "deceased" means the cessation of all brain stem activity. For most organs, this point is too late for the donation to be medically useful; nevertheless, for the adherent to this view, any prior removal would be tantamount to murder. Given the nature of the market for donated organs, the second condition would limit donation to a case where there is a known and ready need for that specific organ. Alternatively, a promise can be made to ensure a proper burial for a donated organ in the event that it is not transplanted. A movement to promote organ donation from Jews to the general population in consonance with halachah has been spearheaded by the Halachic Organ Donor Society.
Most religions like the Roman Catholic Church are in favor of organ donation as acts of charity and as a means of saving a life. Some impose certain restrictions. For example, Jehovah's Witnesses require that organs be drained of any blood, and Muslims require that the donor have provided written consent in advance. A few, such as certain branches of Orthodox Judaism consider it obligatory. However, a few groups disfavor organ transplantation or donation; notably, these include Shinto and those who follow the folk customs of the Gypsies.
Expanding a little more, here is what I found in reference of the Shinto beliefs:
Organ donation in Shinto
The Shinto faith is very much bound up with the idea of purity, and the wholeness of the physical body.
Organ transplantation is comparatively rare in Japan because the body after death is impure according to Shinto tradition.
Shinto traditions also state that interfering with a corpse brings bad luck.
Families are concerned that they might injure the relationship between the dead person and the bereaved (known as the itai) by interfering with the corpse.
This means that many followers of Shinto oppose the taking of organs from those who have just died, and also would refuse an organ transplanted from someone who has died.