Here's where new medical technology could step in to save the day. Bioartificial organs are created by taking healthy cells from the sick organ of a person, and using a scaffold - a synthetic or organic frame to make sure the cells form correctly - to grow a new one. This would solve the rejection problem: the immune system would recognize the replacement as part of the individual, because it is. The technology to complete this process is also fascinating. It's similar to a 3D inkjet printer: the cells are placed, one by one, like drops of ink into the shape of the new organ. This is incubated to encourage it to grow like living tissue, instead of dying, and then can be transplanted just like any other organ. The main researcher for this technology, Dr. Anthony Atala, discussed it in more detail in a TED talk.
I've known several people who became organ donors, and several others who have needed organs. It's a tough process on either end, and often risky for both donor and recipient. Technology like this could help make the process a bit easier, and improve the success rate. It's truly a testament to what medical technology can do to improve our lives.