This was something I used to feel very strongly about when I was younger. Of course back in those days I wasn't as on top of my nutrition as I would eventually be, and I also would sometimes get swept up in the latest hyped concern ("if you don't take a full week off every 12 weeks, you'll overtrain and actually lose muscle!!!")
While I would come to value to practice of backing off a bit, maybe a simple day off possibly with some 'active rest', or even just the mental value of time off, I don't think it's absolutely essential to actually program weeks off (unless of course you're a PLer following a complex, lengthy program, or some crazy periodization).
That's not to say that a week off here and there will hurt the average gym goer, because no matter how much you may be truly convinced the mirror is telling you otherwise, you're not going to lose muscle in a week's time. I mean, as long as you don't go out of your way and do as much destructive stuff to your body as you can in that week -lol.
Unless I'm addressing a nagging injury (ie. my right wrist was killing me a short while ago, so I took 2 days completely off, then resumed, albeit with legs so I would have a 3rd day with no direct stress on my aching joint), my approach is simply to 'go through the motions', but with considerably less weight, and much less volume. My thinking is that I'm still getting blood and nutrients moving through everything, but I'm not actually giving myself stress that requires any real recovery.
I have a feeling we're gonna have some varied approaches in this thread, which is what you should expect. Everyone's recovery ability is different, as is their training intensity, volume, and out of gym/work/school/social stresses from day to day.