I started playing D&D when I was young, circa 1983, and I distinctly remember miniatures being mentioned in the rulebooks, and available to order in the Mail Order Hobby Shop catalogs, but entirely out of my budget. My first DMs didn't use them, so neither did I. As I grew with the game from a child into adulthood, my combats became increasingly more narrative, my players more in tune with my descriptive style. And a clear majority of the players with whom I've gamed over the years (upwards of 80-some-odd individuals by now) were people I introduced to the tabletop gaming experience. So it was never an issue.
I would occasionally use some sort of tactical representation in large, convoluted combat scenarios, but it was largely a visual aid to supplement my narrative. It helped me and my players keep track of their opponents. By the time 3E came around, despite nearly two decades of gaming, I could count the number of combats I'd supplemented in this fashion on my fingers. Keep in mind, I'd even really enjoyed 2E's
Players Option books, and implemented their ideas into my game. Without minis, counter, tokens or combat maps.
It was obvious in the 3E combat chapter that the game was written and developed with tactical aids in mind. With the faith and cooperation of my players, who enjoy my descriptions and ask questions where necessary, I proudly implemented every rule in the book without using a combat grid.
Since my players trust me to provide them with a well-adjudicated and exciting combat experience, I reward them by assuming their characters act with the practice and experience they've garnered. If a PC can move to avoid an attack of opportunity and still reach their target, I'll assume it into their movement. If a PC can't reach a target and attack in the same round, I make sure they understand that they gage the distance appropriately before they act. If it's close, I might inform them that they're uncertain.
That uncertainty is one of the reasons I stay away from tacticals. Although accomplished combatants and tacticians will have a better picture in their heads of any given battle, and I treat them accordingly, the chaos of combat is otherwise prone to make things less clear. I don't feel that the wizard should generally be capable of planting a fireball unerringly in a position where it strikes several enemies and only barely misses his allies. If it's possible, I'll make it possible... but not guaranteed. A combat map offers too many guarantees.
And then there's the matter of running a game that takes place entirely in the imagination, as opposed to one where people focus on minis or counters. I find focusing on the combat map to be distracting, given my style of DMing.
(There's also the matter that combat is not the most important aspect of D&D to our group... Implementation of accessories that slow the narrative while people study a map and move pieces around tend to make battles even longer. Combats are fun, but I don't want them to overshadow the next roleplaying encounter.)
In 4E, much of the tactical power of the game is placed firmly in the hands of the player, with powers that involve small, specific adjustments of position and movement at frequent points throughout the combat. To be fair, this makes the gameplay reliant enough on tactical aids that I am unable to justify running a game without them. I've run a few smaller 4E combats without minis, but that's it. Consequentially, after an initial experiment with the new ruleset, I reverted to 3.5.
Sorry for being so windy.
With gamers who don't approach the game the same way I do, I often find it necessary to defend my decision to run without combat maps. (Though I'll still occasionally use them for epic battles, since pulling out a pretty map and some counters makes it feel like kind of a treat...) And to be honest, I use software that allows me to track every combat on a map on my laptop... so it's not as if I have to guess at anything. I do have players who play in other games that use them... but they very much enjoy my play style as well. And I've enjoyed a level of devotion from my players that many of my fellow DMs often don't seem to.