Any decent i5 or i7 with 16gb ram and at a minimum 970 GPU will run VR alright. Look at Vulcan's system, I know several in DCS that have that same sort of arrangement and they don't complain much about VR performance. If you want max fps most of the time, you will need a faster GPU than a 960, but it still WILL work alright from what I've seen.
I've played AH3 (and other VR sims) with a 6700, 6800, and 6850 CPU along with either a 1070, 1080 (sli, but SLI isn't working well with VR..yet), and a new Titan. No issues in terms of performance, but you do have to manage your expectations in terms of resolution/etc with VR right now IMO. I have Rifts and Vives, and am still waiting for the Razer OSVR unit, but so far both the current VR types seem to perform about the same with similar systems.
IMO I would get at least a 1070 and a decent CPU/MB/Ram platform, if you want to have the best VR experience, again though, you CAN still get a reasonable experience with a 970 and a decent CPU/MB/Ram system, or even slightly lower performing GPU/CPUs. Again, manage your expectations is all in terms of FPS in VR is all.
SirNuke - Rift for cockpit/sim games, all the way. I've had both for a while now, and in every sim/driving game out there the Rift is superior to the Vive. There is a great military FPS/Arma3 like VR game called Onward that only works on the Vive, and that so far is the only game I've found where the Vive is superior, and that's mainly because it's the only option. If you like walk around/room scale non-sim like games, the Vive is a great unit, but again, in sims, the Rift is a lot further along in most games, including AH3, DCS, WT, Elite Dangerous and so on.