I'm getting some great feedback and resistance to my new rule. I love resistance! It keeps me on my toes. As I said before, I'm exploring a better way of explaining what appears to be word-final voicing in PIE based on Jens Rasmussen's published input on an earlier voiced nominative singular *-z and the simultaneous existance of *-d. My view remains that they come from deictic stems *so- and *to-, and I'm sticking to that story. Yet this implies word-final voicing which is exceedingly rare and therefore yucky in a good theory.
Tropylium had got me thinking about a rule based on secondary stress after objecting to the idea that the rule would only occur in unstressed positions, but after pondering on that yesterday (and then again at 3AM tossing and turning), I came back to my senses and realized why this cannot be. As I have it so far, my Reduction rule effectively bifurcates a former schwa (represented as unstressed *a in my MIE notation) to either supershort schwa (the default change) or regular schwa. Since my Syncope rule only deletes supershort schwa but not regular schwa, evidentally the voicing can only have occured before the supershort schwas. This may sound contrived to some but the phonetics are really starting to make sense to me, so let me explain it in detail now.
The reduction or deletion of a word-final vowel with simultaneous gemination of the preceding consonant is commonplace (e.g. Japanese desu /des:/) because this is merely a matter of compensatory lengthening and syncope, both of which are commonplace linguistic processes in languages worldwide. It's important to recognize however that a preceding consonant need not geminate as the following vowel is deleted. In what way the vowel is deleted depends on the idiosyncracies of the language in question and its individual speakers. On the phonetic level, all that's happening in this sort of gemination is a transfer of duration from the vowel to the neighbouring consonant such that the overall duration of the sequence remains the same.
In the rule I'm now proposing for Pre-IE, I suggest that this same gemination occurs, but only in unstressed syllables. Again, this is not objectionable considering the example of Old English wīte "punishment" from wītje (no gemination in accented syllable) versus Old English wēstenne "desert (dat.sg.)" from wēstenje. Compare also wīte < wītje (with long vowel) versus cynne "race (dat.sg.)" < cynje (with short vowel). The importance of the latter comparison will make sense as I explain below. If the lack of stress is some factor in (but mind you, not a *cause* of) this rule, then why and how would this have occurred phonetically?
After pondering, I realized why this might be. In a stressed syllable in MIE, such as in genitives ending in *-ása, I don't expect gemination (and hence later voicing) since the thematic vowel *e in *tesyo "of this" suggests that the *s in the genitive singular was always voiceless (i.e. from early Late IE *təs-ya) in contrast to a voiced *-z in the early nominative singular ending where thematic vowel *o always arose just as it did before other voiced consonants. So clearly in an accented syllable, gemination did not occur, probably because the vowel was stressed and therefore longer in duration. The added duration of the vowel in the stressed syllable would understandably be a credible factor in blocking gemination during Reduction of the word-final supershort schwa (similar to the Old English example above) because a geminated consonant would then compete for time with the accented vowel. The longer stressed vowel naturally won that battle, gemination did not occur, and therefore the genitive remained voiceless. It's brilliant!
In an unstressed position however, there is no such competition for length because there are no tidal effects from the presence of stress. An unstressed vowel is comparatively shorter than a stressed vowel anyways. So as the vowel was reduced at the end, length was then transferred to the preceding consonant to maintain the same overall duration. Thus the unaccented sequence *-Vsa as seen in the MIE nominative singular was free to evolve to *-Vsᵊ (Reduction) > *-Vssᵊ (Gemination) > *-Vzᵊ (Voicing) > *-(V)z (Syncope).
I've also thought of a good reason why secondary stress would be insufficient in explaining this gemination. While secondary stress would have once donned most instances of the agglutinated deictics *-sa and *-ta in MIE, it wouldn't have naturally done so in unstressed MIE *kʷai-ta > *[kʷittᵊ] > PIE *kʷid "what?" yet the inanimate pronominal marker shows voicing nonetheless. Therefore it is the absence, not the presence of stress, that must be a factor in this Gemination rule and this would simply be because a stressed vowel in this language was phonetically long enough to compete against neighbouring consonant gemination in contrast to unstressed vowels which lacked the strength to resist it.
There! Now is this new theoretical account good enough to please the populus?
 Van der Hulst, Word Prosodic Systems in the Languages of Europe (1999), p.342. (see link).
 Blevins, Evolutionary Phonology: The Emergence of Sound Patterns (2004), p.173 (see link): "In many languages, a stressed syllable is longer in duration than a segmentally identical unstressed syllable."
(July 04 2008) I corrected "Since my Syncope rule only reduces supershort schwa but not regular schwa [...]" to "Since my Syncope rule only deletes supershort schwa but not regular schwa [...]". Sorry for the potentially confusing choice of words.