I remember going downtown with my grandmother when I was very young and one thing I recall was people crossing the street from one corner to the oppposite corner at intersections.
I mentioned this to my father when he was still alive and asked how this could be.
The answer was simple. Part of the cycle was red lights for all motor traffic at which time pedestrians could cross in all directions including diagonally.
Was this a common practice in other cities? Does it still occur anywhere?
yes, this is quite common in very large cities, where pedestrians outnumber autos. New York has quite a few, the famous intersection in Shibuya, Tokyo is an example from abroad.
https://www.youtube.com/watch?v=WWbYfBfrHjA
I can't quite remember why they died out in some cities. Denver used to have them all over the place. Many cities are putting them back in. Seattle has more than a few.
The phase is known by several names: pedestrian scramble/scramble intersection, Barnes dance, diagonal crossing, X-crossing.
There's one at 7th & H NW in DC, up the block from the Verizon Center. Extremely high volume of pedestrian traffic due to the arena, a subway station on that corner that serves three lines, a movie theater next door to the arena, and loads of restaurants (most of which are there because of the arena).
The wrinkle at this intersection is that the pedestrian crossing time is not restricted to the "all-way crossing" phase–pedestrians also get a traditional walk sign parallel to traffic. The planners felt pedestrians would cross against the light if this weren't allowed (almost certainly true), and they were concerned that turning traffic would back up the streets waiting for pedestrians. So they banned all turns at that intersection (you can see the "NO TURNS" and no-right- and no-left-turn signs in the Street View image). There are signs as far as a block away warning of the restriction. Doesn't matter, though–plenty of drivers turn anyway. I've seen drivers arguing with cops when the cops are out directing traffic, basically a "who are you to tell me I can't turn" kind of attitude.
The link below may lead to the map instead of Street View due to how the Google Maps app works.
https://goo.gl/maps/aH7tkrPkqhC2