I used the flat d6 weapon damage in Friday's Forsaken Halls session (a report on that later), and...
No problems whatsoever.
I decided to use the 2d6-keep-the-best-d6 idea for 2handed weapons, but none of the players happened to have a 2-handed weapon, so that never got tested.
There were no complaints, and no confusion. Some quick observations:
-It seemed to bring basic combat in line with some "standard" spells like magic missile and cure light wounds, both of which use a basic d6 also (we used Labyrinth Lord for the Halls session just for fun).
-There was a magic javelin (+1) that got a lot of use. I suspect this item may have been eschewed in favor of a non-magical d8 or d10 weapon in variable-damage campaigns due to basic number crunching. Just the impression I got.
-This option may have actually sped up combat just a hair, no more hands hesitating over the dice pile ("what damage does a crossbow do again?"). Everyone pretty much kept a d20 (for attacks and saves) and a d6 (for damage, initiative, and "skills") in easy reach, and saved the other dice for special occasions.
I think a couple of more sessions with non-variable weapon damage are in order to determine if this option is actually better, but I can safely say it definitely wasn't worse, and caused no problems, in case you were thinking of trying it out yourselves.