In every single game I've played in, or run, we've always just sat at the table and talked it out. No minis or grid maps. Maybe I'm old school... but when joining a brand new group, they did the same.
When playing the game without a map, moving "around" things cost more movement than moving in a straight line. If I were to put a "method" on how we verbally do it, it's more like using a bit of string on a map to indicate where you are going. Quite simply, we don't count squares and rarely even feet.. we just say "my movement is X, can I get into position y" and the DM says "Yes/No/Attack of Opportunity/etc".
Every once in a while, we pull out the map or draw a quick grid to work out some heavy tactical stuff.
If we had to switch to 1 square diagonals, our verbiage would no longer match what the game rules dictate. There would be a disconnect between our "real world representation" and the abstraction. Firecubes and "round squares" would no longer match how we describe our characters moving around the physical virtual world.
This wouldn't be a problem, but now I hear that there's rammifications with certain Powers and Abilities that may cause a problem if you don't have a 1 square diagonal.
Maybe we can just play it that changing direction costs more movement, but otherwise a straight diagonal is like shifting the "grid" so that you are basically moving forward? So you can move 1-1-1 in a straight line in any direction, and it costs an extra 1 square whenever you turn?
Basically that would be how we've been playing it in our "fast and loose" descriptions...