Cyborg 009 is one of the earliest pioneers of this, even preceeding X-Men of all things. 001 (infant Ivan Whiskey) is Russian, 002 (gang member Jet Link) is American (specifically, he's a New Yorker), 003 (ballerina Francois Arnoul) is French, 004 (East German escapee Albert Heinrich) is German, 005 (pacifist Geronimo Jr.) is a southwestern Native American, 006 (farm/chef Chang Changku) is Chinese, 007 (Spy Great Britain) is British, 008 (refugee/guerrilla fighter Pyunmar) is African (unspecified where), and 009 (Leader Joe Shimamura) is half Japanese, half American.
Gundam usually has varied nationalities among its characters. Curiously, it's very rare for the main character to be Japanese (even partially).
Mobile Suit Gundam had Amuro (either Canadian or Mexican depending on version of the story), Bright (from Hong Kong... unclear if he's of British or Chinese descent), Ryu Jose (Japanese-Latino), Mirai Yashima and Hayato Kobayashi (both Japanese), as well as a number of Space Colonists.
Victory Gundam had a unusually (at the time of the series) high variety of races amongst the recurring cast. The main character, Usso, is European, but Shakti, the female lead, is of African descent, as are a large number of side characters.
Mobile Suit Gundam Wing's True Companions were Japanese (Heero), United States (Duo), European (Trowa), Arabian (Quatre), and Chinese (Wufei) descendants, though the important nationality distinction for them was that they were all Colonists and not Earthlings.
In Planetes, the Technora Corporation Debris Section has two Japanese astro-trash collectors, a Cuban-American captain, a Russian first-mate, a caucasian (probably American, but at the same time fulfilling Japanese middle manager stereotypes, and relatively harmless) Pointy-Haired Boss, his boot-licking toady from India, and a German Office Lady temp.
In the anime. In the original manga Fee is clearly black and the office staff don't exist.
Planetes also feature at least two mixed marriages: Fee's stay-at-home hubby is white, while Yuri's late wife was Japanese.
In Kagaku Kyujotai Technovoyager, the team consists of a Japanese (who is the central character in the series), an American, an English, a Jamaican and a French. However, the characters' nationalities were changed into American when the series was exported to the US as Thunderbirds 2086.
Black Lagoon — kind of. Revy is a racially Chinese American, Dutch is black American, Benny is Jewish American and Rock is Japanese... Not-american. Defining them as 'good guys' is kind of a stretch too. Since they're pirates, Equal-Opportunity Evil may cover them just as well.
If you expand the scope a little to include some of the people they have happened to have teamed up with on occasion, the list includes the Russian Balalaika, Taiwanese Shenhua and her Irish driver, and the presumably Hong Kong Chinese Mr. Chang.
In the Space Western-cum-Humongous Mecha anime Sei Juushi Bismarck, the main team of heroes consisted of an Englishman, a Japanese, an American and a Frenchwoman; each one wore the appropriate flag on his/her battle helmet and uniform. When the show was Americanized as Saber Rider and the Star Sheriffs, the Japanese and French characters' nationalities were ignored. However, since the show was a Space Western and the American character was a futuristic cowboy, he got to stay American. The Englishman also retained his nationality because one episode's plot was based on his status as a descendant of Scottish royalty.
D.Gray-Manloves this trope. Allen is British, Lenalee is Chinese, Kanda is Japanese, Krory is Romanian, and Miranda is German. It's the same with the bad guys; Tyki is Portuguese, Lulu Bell is French, and Skinn, Jasdero, and Devit are American. Listing all the examples this series has would take too long. Kanda's nationality has never been confirmed, and is always given as 'Japanese?' with a question mark. Recent plot developments suggest he is more likely to have been born and raised in China.
Even the bad guys get in on the action — Cancer is Chinese, Bullhorn is Mexican, Wilder is American, and Hydra and Buster are (East) German. The rest of the Decepticon-allied humans in the series - Clouder, Giga, and Mega - are of unspecified nationality.
The mercenary pilots of Area 88 primarily come from the NATO powers: Americans, West Germans, Brits, and others. The main character is Japanese.
Don't forget the Blade Breakers! Beyblade's main protagonists were Tyson (Japanese, and the leader), Ray (Chinese), Max (American), and Kai (Japanese). They ended up competing against and defeating the national teams of the countries they were originally from (except Japan, which was their team), becoming world champions.
The pilots from RahXephon; it's a surprisingly blatant example.
From The Legend of Koizumi, when time comes to save the world from Space Nazis, we get the main team of Junichiro Koizumi (Japan), Vladimir Putin (Russia), Pope Benedict XVI (The Vatican), George H.W. Bush (U.S.A.) and Yulia Tymoshenko (Ukraine).
The Mighty Bombshells, a failed attempt to bring the American style of comic book art to manga, consisted of Ms. Liberty and Dynamite Girl (U.S.A.), Fireball (England), Blitzkrieg (Germany), Red Dragon (China), and Cherry Blossom (Japan).
Shaman King does this with some of the teams. But since there are really only four factions in the shaman fights, it's natural.
The Idol Choujin in Kinnikuman include, by the end of the series: Kinnikuman and Rikishiman/Wolfman (Japanese, although technically Kinnikuman is an alien), Terryman, Specialman, and Geronimo (American; Geronimo is a full-blooded Apache), Robin Mask and Neptuneman (British), Warsman (Russian), Ramenman (Chinese), Buffaloman (Spanish), Brocken, Jr. (German), and Canadianman (Guess, dummy.)
Sequel series Ultimate Muscle features the New Generation Idol Choujin: Kinniku Mantaro/Kid Muscle (from Planet Kinniku), Terry the Kid/Terry "The Grand" Kenyon (Japanese-American), Kevin Mask (British) Seiuchin/Wally Tusket (Irish), Gazelleman/Dik Dik van Dik (Tanzanian), Checkmate (from Monaco), Illoukhine/Comrade Turbinski (Russian), Jade/Jaeger (German), Scarface/Eskara (Italian), Barrier-Free Man (Swedish), Chaos Avenir (alien from an unknown planet), and Hanzou/Hanzo the Horrible (alien from the planet Iga)
There's also virtually every villain group from either series, but we'll spare you the details.
Gravion Zwei featured the Gran Troopers — Faye (Chinese), Alex (American), Hans (German), Ivan (Russian), and Jose (Brazilian).
Silent Möbius has the AMP team, Kiddy Phenil (Australian), Lebia Maverick (American), Katsumi Liqueur (half-Japanese, grew up in Hawaii), Lum Cheng (Chinese), and Rally Cheyenne (half-human, mother's nationality unknown). Yuki Saiko, Nami Yamigumo, and Mana Isozaki are Japanese.
The superheroes of HeroTV in Tiger & Bunny come from a variety of backgrounds, despite not being an actual team. Kotetsu/Wild Tiger is Japanese. Antonio/Rock Bison is Hispanic or possibly Mexican descent. Ivan/Origami Cyclone is Russian. Pao Lin/Dragon Kid is Chinese. Keith/Sky High is American.
Code Geass has this somewhat with The Black Knights. As they are a Japanese resistance group, the vast majority of its members are, obviously, Japanese. However, its core members include Britannians (Lelouch/Zero, Diethard), Indians (Rakshata), and later, Chinese (Xingke, Tianzi). It's explicitly stated that they are fighting for justice and to rebuild the nation in a new vision as the "United States of Japan". Your ethnic background is of little importance.
Jewelpet Twinkle has the Magic Academy's students. The protagonist Akari is Japanese, Sara is half-Japanese, half-Indian, Miria is from the USA, Leon is Austrian and Nicola is Russian.