Most states require what is called 'Informed Consent'.  Generally speaking, Informed Consent requries the organ, eye and tissue bank to ask for permission for each specific tissue that will be taken and what it will be used for.

For example, the requestor could say 'Do you consent for a blood sample to be taken?  It will be used to scan for diseases as well as to match blood types with a potential recipient'. 

The requestor would go on to ask for consent for the heart, liver, pancreas, intestine, bone, skin, veins, heart valves, eyes and so on.

The NOK can consent for or refuse consent for any organ or tissue.  The NOK could provide consent for a heart but deny consent for veins or vice-versa.

A good example of choosing which tissues to donate is the heart.  Some belive that the heart holds a person's soul.  They may consent for bones, skin, veins, eyes and so on but deny consent for the heart so that it may be buried with their loved on.

The bottom line is that everything is up to the NOK.  Donation is not a blanket 'yes' statement; there is always specific power given to choose what is to be donated.