What is the current status of research that might tell more about
predicting which antidepressants work best for which patients and why?

I've heard about the research on depression that says that
antidepressants are (roughly) fully effective for a third of patients,
partly effective for another third, and not effective at all for the
remaining third. I've heard that trying a second antidepressant can
often be helpful. But there are some patients who been have tried on
multiple (sometimes ten or more) antidepressants, which were either
never effective to begin with, or pooped out within a few months.

I'm interested in theories and studies on both why some depressed
people respond well to antidepressants whereas others don't, and why
different antidepressants (particularly of the same class, such as
SSRIs) work for some people and not for others. For example, for
patient A, antidepressant X works but antidepressant Y of the same
class doesn't, but for patient B, Y works and X doesn't. I'm fairly up
on the usual antidepressant models: synaptic cleft between neurons,
neurotransmitter reuptake, basic brain anatomy. But I get a sense that
researchers don't yet have much of a clue about WHY different
antidepressants affect people differently. Is that the case? Are they
making much progress?

Thank you.

Here is an interesting study regarding nonresponse to fluoxetine (Prozac):

http://ajp.psychiatryonline.org/cgi/content/full/163/1/73

The answer to your question is probably unknown.  This is an active
area of research, so there's quite a bit of literature that continues
to come out.  I'm not a psychiatrist, so this isn't my area of
expertise, but here's what I know...
There's quite a bit of research suggesting that different
antidepressants act differently.  Although the newer ones are named
'selective' serotonin reuptake inhibitors, the reality is that many of
them interact with other neurotransmitter systems, and these
interactions may cause a different clinical response in any given
patient.  Another emerging area of thought is that the effects of
ssri's (or older antidepressants like MAOI's, tricyclics, etc) may be
due to the fact that their regular use causes down-regulation of the
receptors at the synapse itself - so the drug's effect may not be due
so much to the actual ability to block serotonin reuptake, but to the
cellular response to the medication.  Personally, i like that
explanation because it could explain a mechanism for the clinically
observed delay in response when starting antidepressants in patients. 
Finally, there's research emerging that suggests that the serotonin
receptor is more diverse than we've thought.  It's known there are
multiple types/categories, but some research suggests that certain
polymorphisms (subtle genetic variations that exist in the population)
exhibit different clinical responses to antidepressants.
So, like i said, nobody knows for sure.  But these are the directions
the research is heading.
try searching on pubmed.com for some of the above topics...here's one
about gene polymorphisms.
http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=pubmed&dopt=Abstract&list_uids=16351675&query_hl=2&itool=pubmed_docsum

It is important that you consult your doctor on all antidepressant
medications and let them know of any developments in treatment,
including side effects. Your doctor will prescribe treatment based on
the pattern of your depression, its severity, persistence of symptoms
and history.

The doctor will also consider if the patient has another psychiatric
conditions such as an anxiety disorder or other medical problems such
as cardiac problems. The doctor will consider how likely it is that
the patient will follow directions such as staying on a restricted
diet when a medication may interact with food.

http://www.depression-guide.com/antidepressant.htm