If I were still using a 1" = 5' scale and miniatures, it would be diagonal movement counting 1-2-1-2 all the way. But as it stands, I discovered lately that I have no cause to use a scale that fine. All of the movement rates, missile ranges, and spell areas in my edition of choice -- OD&D -- are nearly always measured in 10s of feet, not 5s of feet, and the game itself assumes a square grid where 1" = 10'.
I can mark the positions of characters and monsters using pipped d6s of varying colors, (the numbered ones are for rolling, thank you very much!), I no longer have to cart around loads of minis and chessmen, and best of all, my battlefield just quadrupled in size. I can have huge wilderness battles, or I can fit four times as much dungeon on the matt without erasing the marker.
How does diagonal movement fit into this? I decided that since the system was abstract enough -- lots of characters can fit in a 10' square and fight it out -- the best rule would be that at very close range (i.e., adjacent squares), no moving or attacking into diagonal squares. Beyond that, though, it alternates in order to satisfy everybody's sense of distance and realism. Effectively, since I've switched to a 10' scale, diagonal movement and distances count 2-1-2-1 instead of 1-2-1-2. It's wierd how minor changes like that can actually improve the experience of playing tactical battles on the tabletop, but it did. *shrug*