I can give you some real-world experience with this.
Until last week, I played WoW for years on a single 8800GT video card (512MB), with an Opteron 165 processor (dual core) and 2 GB of RAM. OS is WinXP. With that setup, which is 5 years old except for the video card, I was able to play WoW almost seamlessly at the maximum allowable settings, on a 1920x1200 monitor. The only area I ever had stuttering was Dalaran, and that was only after arriving at the Inn after using my Hearthstone which is set there. It took a few seconds for the objects to all load; then it was back to smooth as butter. In no other situation did framerates ever dip to a point where I noticed stuttering; not in instances or anywhere else. I never actively monitored the framerates, but they certainly appeared adequate at all times (except the Dalaran exclusion above).
Now, my new computer is sporting a Core i7, 6GB RAM, and two 4890 video cards in Crossfire. Running Windows 7 RC. It's light years beyond my last computer, but it has not noticeably improved my WoW experience since it was already about perfect the first time around. Also, this new build did nothing to fix the initial Dalaran stutters so I am wondering if maybe it's a network / connection issue. My internet speed is 10M, so it's not on my end. (But I could be wrong... I don't pretend to understand the intricasies of the internet...)
So with that all being said... You certainly don't need high-end to get the most out of WoW, since I was running fine at max settings on high resolution with a simple 8800GT and a 5-year old dual core. Any modern-day equivalent to that would likely provide you with great results, and FAR better results than you are used to.