I think most of us would displace to pick up a river, resource, or other goodie. Those of us who favor CxxC realize that it's very difficult to come up with schemes to work 12 tiles for each city that way, so an excuse to loosen up a little, at least in the core, is welcome.
In the farmlands, it's a different story. You really don't care a whole lot about freshwater, just spam-slam in the cities as close as you can to get those beaker/gold harvests going. In the intermediate areas, I'll still displace most of the time.
City placement's an art. I don't pretend to have it solved.
kk
Not long ago I used to be in the same boat as you, Shad, but I've come to like the CxxC grouping a LOT. I will admit, though, that my city layout is not even close to rigid. My spacing is, on average, something more like CxxCxxxC. My main goal is that each city is at least CxxC from two other cities, such that I can move units between them in a chain all the way through the early IA. I tend to leave bigger gaps around hard-to-work or non-growth terrain like desert, jungles and mountains (which, once I clear away the jungle, I will try and fill in some). I try to get my cities on fresh water whenever possible early, but relax on that a bit once aqueducts become available.
In short, Shad, the main reason for the CxxC spacing is to have a strong growth and defensive potential in the AA, where it is most important. It will snowball into the MA and IA and make your later game go a lot better. As stated before, your cities cannot grow above 12 until sanitation, and as for me (and most advanced players) the beeline to replaceable parts is FAR more essential than medicine -> sanitation and will suck up dozens more turns where CxxC retains its usefulness. So why not use the closer spacing before then and take advantage of every spot of terrain you can??

As proof-of-concept, before I started CxxC spacing, regent was a tough gig. Now I compete at Monarchy level often without struggling at all (last night I netted a lousy plains/hills/no-fresh-water/boxed-in-by-four-AIs start with the Persians, but still pulled out a 1000AD conquer VC) I still have yet to have a happy Emporer game, but that has more to do with my refusal to micro-manage
EDIT: Ok, to be fair (and before you pros pick on me), the close proxy of the AI probably helped my early VC with the Persians, as I was able to secure iron and spam Immortals out extremely early. Since I had close borders, the immos were easy to get to the front, and I had wiped out all the civs on my continent before 0BC. This allowed me to REx out, get some boats built, and go after the other continent (which, being smaller and with fewer civs, and never having met anyone on my continent before I destroyed them all, was extremely tech backwards). In short, its all about knowing and playing the strengths of the map and your civ, too, something that is hard to teach and must be learned through experience.