Hi Dave,
I have owned and used Tokina, Sigma and Tamron lenses on a D90 in the past and never had any issues. Same with my current bodies (see signature) and a few different third-party lenses, so I am not sure what the issue would be.
All three of these make some good and some not-so-good lenses. In the relatively affordable category, say, below $1000, Nikon's lenses do not necessarily stand out and I've often (though not always) seen tests suggesting that others' lenses perform better, but this is not consistent in favor of any particular make. In other words, I don't see that Tamron consistently beats out the other two.
In the >$1,000 category, Nikon often wins comparative tests, though the margins between the competitors get narrower and there tends to be a price difference that may make third-party lenses attractive even when the corresponding Nikon is a hair better. Sigma and Tamron are both catching up all the time, and some of their recent lenses are excellent. I'll be happy to get more specific if you are eyeing specific lenses or focal lengths.
As far as the 'holy trinity' goes, this is a generic term generally used for a combination of wide-angle, standard and tele zooms, each with a maximum aperture of f/2.8. For Nikon, that's usually the 14-24, 24-70 and 70-200. Here again, Tamron has a good offering but IMHO does not yet beat Nikon on image quality. They offer attractive pricing, though. Assuming you are eyeing a holy trinity, I do not see a reason to buy only one make. Mixing and matching seems perfectly fine. If I were to make that step (these lenses are too heavy for my travels, so I won't), I'd probably go for Nikon (wide) and Tamron (standard + tele), but that's just my take here and others may see this differently.