FOXHOUND, Dead Cell, Cobra Unit and the Beauty and the Beast unit from Metal Gear Solid 1, 2, 3 and 4, respectively as well as the Winds of Destruction from Metal Gear Rising: Revengeance, are rare villainous examples of these, even though the first two were technically American military units.
Team Peregrine Falcon/Sparrow from the Metal Slug games: Marco Rossi is racially Italian American, Tarma Roving is British, Trevor Spacey is Korean, Fiolina Germi is Italian (from Genoa), Eri Kasamoto is Japanese and Nadia Cassel is French.
RED and BLU from Team Fortress 2 are staffed by a Bostonian, a Texan, a Midwesterner, a German, a Scotsman (indeed, a Black Scottish Cyclops), an Australian, a Russian, a Frenchman, and... the Pyro.
The Jagged Alliance games allow players to hire mercenaries from around the world. Subverted and possibly deconstructed, however, in that some mercs will not work with other mercs specifically because of their nationalities. Just two examples out of many, Steroid (Polish) won't work with the Russian mercs due to the two countries'... shall we say "less than stellar" relationship, and Scope (British) and Micky (Irish) won't get along either, for reasons you can probably guess. The negative effects from this can be as minor as lower morale, to as bad as mercs refusing future contracts to even quitting your team on the spot.
Despite the protagonists of Left 4 Dead being all pretty obviously American, they represent different slices of the population — a young black professional; a young, somewhat ethnic-looking white female college student; a not-so-young white biker dude; and an aging white military vet. Justified, in that that lineup is pretty much your average Zombie Apocalypse film Four-survior band. And Left 4 Dead is essentially an interactive zombie film.
Left 4 Dead 2 takes place in the deep South, and features characters that, like the first game, portray different slices of life so that almost ever player ends up identifying with at least one character. The survivors include a black, middle-aged, and overweight high-school football coach, a white, young, redneck auto-mechanic, a black female news anchor, and a white escaped convict.
X-COM recruits from around the world, with recruit names randomly drawn from British/American (hard to distinguish), German, French, Japanese, and Russian pools. The soldiers themselves all suck about equally.
In the sequel, Terror From The Deep, the Japanese names are changed to Spanish. They are probably easier to distinguish by the European/American players.
The Firaxis remake has this as well, also adding the soldier's national flag on the uniform. However, the game allows you to change the appearance and name of any soldier, so there's nothing preventing a man with a Japanese name to look like a Scary Black Man. Furthermore, all soldiers have standard American accents despite their nationality. Note: this is from a company whose latest big hit is noted for authentic languages. The Updated Re-release, Enemy Within, does offer more than just standard American accents, which some contain Spanish, French, Russian.
Task Force 141 in Modern Warfare 2 (and the proto-141 seen at the end of CoD4) in the is made up of US, British, Canadian and Australian specops troops (befitting how such a unit would probably be made up if simply due to population, about 90% of the troops seem to be British or American, you have to look hard for the Canadians and Aussies). A horrible subversion occurs in Modern Warfare 2 when it is revealed that The American General in charge of the unit was using it as part of a plan that included the orchestrating of a major war that he would emerge from as a hero. And this plan includes murdering most of TF141, so that there will be no "loose ends".
In Mass Effect 1, Shepard's squad consists of two humans, a turian, a quarian, a krogan and an asari.
In the second game, it's bumped up to three humans (Five with DLC), a turian, a salarian, an asari, a quarian, a krogan, a drell and a geth
The third game reduces the number of available squadmates for a better focus. You now have two humans, a turian, an asari, a quarian, an AI (non-geth), and (with a DLC) a Prothean.
The Psychonauts seem to be this: of the four adults seen in the game, one is German and the other is Brazilian. It's a bit harder to tell since the main cast is made of students, all of whom seem to be American (except for the Canadian Chops). Plus the whole technicolor skintones thing. And there's the Russian kid.
In the Silent Storm series, the Allied and Axis special forces unit are manned by soldiers from Allied/Axis countries, with some from Allied/Axis-linked countries with a few joining them due to neutrality/"own country being a douche in standing up for itself"/"I think the Allies/Axis forces can help my own country be free" issues. One of them is a British from Northern Ireland who joined the Axis special forces unit due to her father's execution by the British as an IRA rebel.
In Silent Storm Sentinels, you get to choose from a combination of the soldiers from both sides of the previous game, since the war is over by that time.
In Hammer and Sickle, you recruit squad members as you go, and they are also multinational, as you travel all over Europe.
In the H-game Dyogrammaton, DAKT is part of a UN response to an alien invasion, with team members from the United States, Japan, Italy, China, and Russia.
Though it takes place in a One World Order setting and there are only accents to go by, the Pillar of Autumn in Halo has what sound like American, Mexican, English, Scottish and Australian soldiers aboard. Presumably non-English localizations also have a variety of accents present.
The advertising campaign for ODST was set (and filmed on location) in Serbia.
Dragon Age: Origins has your NPCs representing all the different races and power factions present in and around Ferelden: Alistair (the Templars and nobility), Wynne (the Circle), Sten (the Qunari), Oghren (the dwarves), Zevran (the elves), Shale (a golem), Leliana (the Chantry perspective) and, finally, Morrigan, who's just out for herself. Leliana is also Orlesian, and Zevran is from Antiva.
The Dragon Age: Origins – Awakening expansion has Anders (see below), Justice (a spirit of the Fade), Nathaniel (a disgraced Fereldan noble), Oghren (see above), Sigrun (a casteless dwarf), and Velanna (a Dalish elf).
Dragon Age II has you recruit team members in Kirkwall, which is overfilled with refugees from the Blight. You have Hawke's sibling (a Fereldan with roots in Kirkwall), Anders (a Fereldan mage with roots in the Anderfels), Aveline (a Fereldan warrior whose father was Orlesian), Fenris (an elf from Tevinter), Isabela (a Rivaini pirate), Merrill (a Dalish elf), Sebastian (a Chantry brother from Starkhaven), and Varric (a dwarf). The "Mark of the Assassin" DLC also adds a temporary companion in the form of Tallis (a Qunari elf).
Fallout Tactics: Brotherhood of Steel allows you to recruit squadmates from a number of races from all over the wasteland after completing certain objectives. This includes humans, super mutants (large green humanoids mutated by a virus), ghouls (radiation-mutated humans, deathclaws (massive beasts with large claws), dogs, and humanoid robots.
In Star Wars: The Old Republic the companions the player has depends on the class they choose. Usually its two humans, two or three aliens and a droid.
Dungeons & Dragons-based video games habitually have your Player Party made up of people from any number of species (usually the core races — human, elf, half-elf, dwarf, halfling, gnome, or half-orc — but more recent titles have gotten more creative). Special mention goes to the Icewind Dale series and Neverwinter Nights 2: Storm of Zehir, which allowed you to hand-craft your own party. In the latter a lore-minded player might create a drow warlock from Ched Nasad, a human fighter from Damara, a fire genasi wizard from Unther, or a halfling paladin from Luiren, with whatever justification they feel fits the bill.
All three factions are in Command & Conquer Free to Play a.k.a Generals 2, this, including the GLA. The European Union is naturally made up of member European states, the Asian-Pacific Alliance are made up of several Asian countries lead by China, and the Global Liberation Army's members come from Third-World conflict zones or just about anywhere, turning them more into Western Terrorists and thus, global (they still have mid-eastern members though). Here's an explanation about the GLA's change into a more diverse faction.