The range of a 3e fireball cast by a 5th level Wizard is 700 ft. (Yes, I know they've probably nerfed that in 4e. I'm using this by way of example.) That's 140 squares, of course.
A party is travelling along a road that runs North - South, and the DM has aligned his battlemat accordingly. In the distance, they see a band of Orcs, at just over 800 feet away.
Under the new rules, the Orcs are either well out of range of the fireball or well within range, depending on whether they are to the north of the party or to the north-west.
(In fact, per the new rules, the new 'diagonal' range of that fireball is 985 feet.)
These new rules are a spectacularly bad idea.