Roll for Initiative?

It's just an extra die roll!
Initiative was not a part of my initial White Box (original D&D) play in 1977-78 and I am still reluctant to use it when I referee. The Little Brown Books do not spell out how to do initiative and although they reference Chainmail for additional rules, I didn't have that item until years later. The result is that I have gotten through many games without using "initiative rolls". When acting as a referee I frequently allow players to roll their attack dice first, before any monsters, but with the understanding that the monsters will take their attack back, even if the player characters kill them that turn. Of course sometimes the monsters "forget" to attack as they die thereby giving me a bit more control over character survival.
Having come from a "wargames" background, I am familiar with "simultaneous" turns. The practice of using "written orders" for each unit and simultaneous movement and attacking was popular in tabletop miniatures wargaming during the 1970's and although we were not writing orders in D&D, the simultaneity of action still seems to make some sense. When a friend came to the table with a copy of Holmes Basic around 1978, I was introduced to the use of Dexterity scores to determine who acted first, and so forth, during a combat round. Generally, my monsters didn't have "dexterity" scores and rolling them up during an exciting combat sequence seemed awkward (and slows down the game). "Get on with the fighting!", I say. (After-all, doesn't the term "melee" imply a swirling hack-n-slash close quarters fight where each combatant takes advantage of whatever opening they are given to score a hit on an opponent?)
The structured nature of later "initiative systems" has always seemed needless (and often dull) to me. But, "Roll for initiative" has become synonymous with "it's time for combat". Many systems have each player roll for individual initiative each turn, thereby slowing the game down and adding to the bookkeeping of each combat.
I prefer combat to be fast and a little chaotic, which actually seems more real to me. As referee, I also prefer to have the freedom to structure each encounter and combat round according to what seems to make the most sense, and to add to the sense of game drama. Allow those with longer weapons such as pole arms to set against the charging enemy and therefore strike first that turn! It adds to the excitement and believably because it makes sense and also rewards tactical play!

When I do use initiative, I prefer something other than "roll a die and add your dexterity bonus". Shadow of the Demon Lord, an excellent d20 game by Robert Schwalb, uses a creative mechanic for determining who goes first during combat. In the Shadow... system, players decide on either a "fast action" or a "slow action". Fast actions go before slow actions and include characters who just strike at their opponent without moving and some spells. Slow actions involve combining movement with attacking and also the casting of certain slower magics. The players must commit each character to either fast or slow action prior to any actions being taken that turn.
My approach to adventure gaming has always been to question everything and whenever possible to improve on the system as written. I realize this is a preference not everyone shares. It is one of the reasons the original rules and the OSR appeals to me. Of course, as the saying goes, your mileage may vary.