Hi gman,
I wanted to look at this question from a slightly different angle. You are half right when you say that we should have bunches of neutral body parts. Your problem is that you are looking at a very high level of organization - organs, physiological structures, etc. Now it's pretty clear that, at this level of organization, organisms are generally pretty well adapted and sleek. The selective forces that maintain a well-organized eye are quite strong, so we generally don't find new physiological structures appearing in the eye that don't carry out a useful function.
If you are willing to look at a lower level of organization i.e. the molecular level, you will find exactly what you predict: huge numbers of neutral traits. To take an example, consider the human mitochondrion. Everybody needs a mitochondrion, and all of the genes it carries are essential to human life. So you don't find "bunches of neutral genes" or anything like that just appearing in the mitochondrion. But if you compare the mitochondrial DNA sequence of a sample of human beings, you will find a huge number of neutral polymorphisms. See
http://www.mitomap.org/cgi-bin/mitomap/tbl7gen.pl for example. These are the "bunches of neutral parts" that you predict. You see, even in a gene of essential function, that is under strong selection not to change too much, you find a huge amount of neutral variation.
You probably won't find this satisfying, because you specifically asked about neutral body parts. But biologists recognise that selection acts in different ways at different levels of organization. By only considering "irreducably complex" systems, which I take to mean coadapted physiological systems, you are only looking at half the story. Neutral evolution is occuring at the molecular level, and it might only surface in the phenotype once those neutral polymorphisms take on an adaptive function under changing environmental conditions.
Cheers
Mick