So, where do I get these magical parts, anyway?

That, dear reader, is a very good question, and one that caused me to have to do a CONSIDERABLE amount of research.  In the end, I only found one vendor who had all of the required components.  The caps I found are made by Nippon Industrial Corporation (NIC).  You will need the following parts:

NRSK102M6.3v8x11.5TRF (1000 µF, 6.3v) (need 6)
NRSK152M6.3v8x20TRF  (1500 µF, 6.3v) (need 6)
NRSK222M10v10x23TRF (2200 µF, 10v) (need 4)
NRSK332M6.3v10x23TRF(3300 µf, 6.3v) (need 6)

I was able to aquire mine directly from NIC as engineering samples, and I received ten of each.  However, it is unlikely you will be able to obtain them this way.  You should be able to purchase them through Arrow Electronics, Hamilton-Hallmark Electronics, Future Electronics, Jaco Electronics, Gateway  Electronic Components, or Gothic Components.  You can also go to their web site at http://www.niccomp.com/worldrep_dist.html-ssi to find a distributor near you. When I initially did this, the parts were brand new to the market and only Arrow even had them in their catalog, and they indicated they were a special order item with a minimum order of 200 pieces of each part.  However, things should have changed since then.  Expect to pay something in the area of $20 for the replacement parts and shipping, plus about two hours of your time.

If you decide to do this, please leave some feedback in this thread in the forums and let us know how you get on!

Credits 

Many thanks to Daniel ~.  Without the sacrifice of his board under the scalpel, this article would not have happened.  I'll let him speak to the efficacy of the repair.