I'm arriving late to the holiday Moire party but I essentially agree with what everyone else has been saying plus a few thoughts.
I shoot with a D800E although I've also used the 800 and 810. A seriously sharp lens (either of the Sigma art primes, the Zeiss apo sonnar, the Otus) will definitely allow moire to be seen by nature of their exceptional resolution. Unfortunately I can't say the D810 solves the issue either: I've seen moire with a D810 and both a Sigma 50 art and an Otus 85, so it's not exempt. Moire often can also be tricky - at one particular combination of aperture, lighting direction, lens selection it occurs but a slightly different angle, aperture, etc and it can be mitigated - at least in situations where you're not at the absolute sharpest aperture of the lens.
The solutions are generally to try stopping down first - I see F/11 as sort of a "moire filter" - doesn't always take care of all of it, but it helps and if I'm shooting a particularly tricky piece of fabric in the studio (say a mesh of some sort), I'll take off the Sigma 35 or 50 art and use the Nikon 24-70 as a replacement. The 24-70 is still a sharp lens, but not in the category of the two Sigma primes (which in the subject distances associated with studio work are about as good as it gets), and that slight drop in resolution usually solves, or at least, significantly reduces the problem.
I should note I don't use Adobe raw conversion products, since raw conversion does play an impact.
At the end of the day I think it's a case of learning ones tools. I shoot in the studio quite a bit so by now I kind of instinctively "know" which lenses and fabric combinations are likely to give me moire problems and I'll use one of the methods described (stopping down, lens change) when I am going to be using them.
-m