However my busy mind also can't resist exploring a new perspective. We can at least say that while the roots Phoenix identifies do indeed appear to "look alike" (although this is assessment is a little too subjective for my tastes), there is no clear pattern in voicing or devoicing that we can immediately ascertain:
#1 *pieh₂- ~ *bʰeiH-Yet what if there is actually something behind these pairs? For the sake of brainstorming, the third and fifth pairs in Phoenix's list appear to suggest a possible solution that might be enticing for future examination. In reconstructed Proto-Indo-European, just like in any living language, there appear to be phonotactic pressures which determined how sounds could validly fit within a word.
#2 *pleh₂g- ~ *bʰleh₂g-
#3 *trep- ~ *strebʰ-
#4 *treup- ~ *dʰreubʰ-
#5 *terp- ~ *(s)dʰerbʰ-
One important curiosity whose origins I have yet to solve to my satisfaction is that there appears to be some sort of "voicing harmony" such that voiced stops (ie. voiced aspirated stops, traditionally speaking) can only appear in a verb root with other voiced stops while voiceless stops can only appear with other voiceless stops. Creaky stops (ie. traditional plain voiced stops), no doubt descendants of ejectives, could occur with both voiced and voiceless stops but not with other creaky stops. So *tep- or *ped- are valid root shapes yet **dʰep- or **deg- are not. Putting aside how this pattern arose in the first place, I wonder if this well-known phonotactic constraint in unison with the presence or absence of the so-called mobile *s-prefix could be to blame for these apparent pairs above.
Here's how I imagine this could work. We start with a root like *dʰerbʰ- which has an *s-marked variant, *sterbʰ-. Following *s in an onset cluster, the voicing of a subsequent stop is neutralized as in English but otherwise the phonotactic constraint I mention above is unaffected. Now what if PIE speakers begin to drop the *s in *sterbʰ-? I'd imagine that, if the phonological constraint still holds, a resulting form *terbʰ- might be pressured into becoming *terp- to maintain the stop voicing harmony of yore. This then could especially help to explain the fourth stem pair in this list: *treup- ~ *dʰreubʰ- (via an *s-marked intermediate variant of *dʰreubʰ-, namely **streubʰ-).
 Denis Sinor, The Uralic languages: description, history, and foreign influences. Handbuch der Orientalistik, v.8 (1988), p.274: "Relating gradation to Uralic or even Finno-Ugric has been criticised because it is only a feature of the languages mentioned above and is not found in any other Uralic language. Scholars have therefore seen gradation in Balto-Finnic and Lapp as the result of parallel, but separate development to the gradation in Samoyed." (see link).