The "three people" format is alive and well here in the US. In my experience, the three are typically defined as/by:
- religious figures ("a priest, a rabbi, and an imam walk into a bar")
- professions ("a physicist, a mathematician, and an engineer")
- racial/ethnic stereotypes ("a white guy, a Black guy, and a Mexican") - note that AmE very strongly implies that the Mexican is also a guy here
- hair color ("a blonde, a brunette, and a redhead") - note that referring to someone by their hair color alone ("a blonde") universally refers to a woman (vs. using "a blond man")
- state of residence ("a Minnesotan, an Iowan, and a Wisconsinite")
It's not unheard of for the setup to use countries of origin, with Poland (IME) usually being the butt of the joke, but those are (again, IME) much less common than the other setups.
It is worth noting that these jokes do seem to be falling out of favor, especially the ones that rely heavily on stereotypes (the example in Peter Jennings' answer about a physicist, mathematician, and engineer measuring the height of a building would be fine in any circle I can think of). They can come off quite well, but they're more than a bit risky. It's probably safer to avoid such jokes, but flipping the script and having the presumed butt succeed because of their stereotype (possibly by intentionally playing it up) is often a good option.
Peter Jennings' answer's joke, in case it gets deleted/lost:
A physicist, a mathematician and an engineer were were each given a theodolite and told to measure the height of Senate House (for many years the tallest building in London, belonging to the University). The Mathematician, knowing his trigonometry, sets up the theodolite, triangulates the building and calculates the height. The Physicist, not being quite so good at maths, swings the theodolite as a pendulum from the top and calculates the height from the time period. The engineer goes down to the Public Records Office and looks up the height, pawns the theodolite and buys drinks for all of them with the proceeds.
It's also not terribly uncommon to mix groups: Peter Jennings' joke could be re-done as "a physicist, a mathematician, and a blonde", with the blonde (who are stereotyped as being some combination of ditzy, air-heads, and dumb; Legally Blonde plays with this pretty well IMO) having the clever idea to go look up the height.
Source: I was born and have lived wholly in the US, mostly in Minnesota.