Not truly averted, just not nearly as obvious (partially to show how humanity is so united, and what not, not to mention the increase of aliens in the crew). In Star Trek: The Next Generation, for example, there was a Frenchman (Picard), a few Americans (Riker, Crusher), an African (La Forge), a Belarussian-by-adoption (Worf, who represents, obviously, a whole other "nation" as well) and an Irishman (O'Brien). Deep Space Nine had an American (Sisko), the Belarussian-by-adoption (Worf), the Irishman (O'Brien) and somebody clearly of a multi-ethnic background (Bashir). Voyager wasn't as diverse but tried to make up for it with more aliens. Enterprise had a few Americans (Archer and Trip), an Englishman (Reed) and a Japanese (Sato).
Stargate Atlantis: The Atlantis expedition is an international team with members from a large variety of countries. You can have a bit of fun spotting the flags on the base uniforms too. Regulars have included a number of Americans, a Canadian (Rodney McKay), a Scot (Carson Beckett — no, I don't know why he has the Saltire on there when Scotland is not an independent state) and a Czech (Radek Zelenka). Extras have included South Africans, Russians, Chinese and a Belgian. And Rodney fangirl Miko Kusanagi is Japanese. Unfortunately, the only Australian was also imaginary.
Power Rangers, at least in its earlier seasons. The episode "A Season to Remember" features a plot in which the villains tried to use their ethnic diversity against them by sparking a fierce Christmas/Kwaanza/Hanukkah rivalry. This episode, incidentally, pointed out the fact there were no Jewish Rangers until Power Rangers S.P.D., where Bridge is Jewish. The Jewish character in that Zeo episode was their geeky ally. It's weird when you consider that Haim Saban and Shuki Levy, who created the franchise, are both Israeli...
Well it is a kind of Multinational Team but most rangers have had American accents,rather than a clean one from Japan,one from America,one from Egypt...
Battle Fever J, the third Super Sentai show had a strange variation of this idea where the members except for Miss America were "from" nations around the world, but since they were played by Japanese actors, they were Japanese who had been sent to Russia or France to be trained in combat and in acquiring tastes for caviar and escargo. The first Miss America was the only exception since she was played by a Japanese-American woman. The peculiarities of this series have something to do with the leftovers of the original idea to make the show "Captain Japan", a Japanese interpretation of Captain America under license from Marvel Comics.
A few other sentai seasons have a Chinese character or a Japanese-American character.
UFO (1970-1). SHADO is a multinational alien-fighting organisation based in England, but commanded by an American (e.g. a British Series aiming for a US market). Other characters are Australian, Eastern European, Indian and Afro-Caribbean. A lampshading of this is the food dispenser on Moonbase, with sections marked "American Meal", "Russian Meal" and so on.
Defying Gravity had four countries represented among eight shipmates. At least one more country was represented among ground crew.
Crew: American (Maddux Donner, Paula Morales [Hispanic-American from Texas], Steve Wassenfelder, Ted Shaw), Canadian (Jen Crane, Zoe Barnes), German (Nadia Schilling), Israeli (Evram Mintz). Most of the ground crew is American, although Dr. Claire Dereux is Canadian and Ajay Sharma (who was originally supposed to go on the Antares) is Indian.
Danger 5, an action comedy about a team of five assembled to assassinate Hitler and thwart Nazi plans in the 1960s. The five are Jackson (American), Claire (British), Tucker (Australian), Ilsa (Russian), and Pierre ('European').