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Historically, the Jewish mob is probably the closest to what you're looking for. Meyer Lansky, Bugsy Siegel, Arnold Rothstein, all mainly known for their connection to early Italian mafia outfits but they were in somewhat ethnic jewish gangs in their own right. There's a problem of differentiation because they generally were connected to Italian mob outfits, and the larger scale criminal endeavors were joint endeavors. Similar to nowadays with the Mexican mafia using central American gangs for certain jobs. Early American crime wasn't based on drug trafficking and before prohibition was primarily centered around gambling (gambling parlors as well as local lottery pools) and paying for protection. Meyer Lansky in his early years was well known for his gambling parlor, specifically it was known to be less likely to be rigged (he relied on more modern understanding of odds) and Rothstein was famously believed to have rigged the world series.

Another group of famous ethnic gangs are the collection of tong and triad found in San Francisco's Barbary coast/Chinatown. It's tricky to discuss because membership wasn't as regimented, a member of one tong could be a member of up to around 6. This made gang conflicts much trickier. Now in San Francisco (and portland) the main criminal organizations were focused on roughly the same as the Italian and jewish, but with a much weaker focus on sports betting. Opium smuggling/parlors and prostitution were very much a point of wealth. Particularly women were essentially treated as a commodity, around 1880s-1910s (the exact time escapes me) the Chinese male to female ratio was often close to 20:1 due to immigration policies of the day against the Chinese (Chinese exclusion act and the geary act) . This led to a pretty large black market for Chinese women, often a woman could be held as a sorta bank note between rival tong during war/deals. There were severe ramifications of a Chinese man bedding a white woman, even a prostitute, due to the craze of the yellow peril.

Sources:

Herbert Astbury's the Barbary coast.

Foundsf.org - a collection of archival news articles.

Hatchet men: the story of the tong wars by Richard dillon

NOTE: I'm not a historian so I can't tell you whether the sources are considered_good_ sources by the historian community. But they seemed generally well researched in my amateur view. I believe Hatchet men is primarily sourced from historic police files (and all the biases contained within). I used them to write a screenplay a couple years ago about a turn of the century gang war.

The reason traditional organized crime groups tend to be from certain ethnic backgrounds (Irish, Italian, Russian, Jewish) is because these groups comprised of immigrants who were considered outsiders from mainstream American society (i.e. "non-white") during the time when most of them immigrated (1800's). The immigrants tended to be poor and lived in slums, most notably in New York City.

Since these immigrants were considered outsiders, many of them tended to band together to illegally make money since it was hard for them to find work at a decent wage due to their ethnic background. New York City street gangs were primarily dominated by Jewish and Irish organized crime until the influx of immigrants from Southern Italy and Sicily from the 1880's onward. The immigrants from Southern Italy and Sicily were so used to thousands of years of oppression and poverty and tended to only trust people from their immediate family and people from their town. The early founders of the Bonanno Family in New York is a good example of this because they tended to be mostly from Castellammare del Golfo, Sicily and didn't trust people who were from other parts of Sicily.

After Italy unified in the 1860's much of Southern Italy and Sicily was left unguarded from police and the government for many years, which led to the creation and rise of the Mafia in the vacuum. The Mafia is unique because of how much family blood ties, culture, and distrust of government permeates throughout the culture. When the immigrants through Southern Italy and Sicily arrived in the US, they soon became the dominant criminal group. Where NYC had been controlled by the Irish and Jewish gangs the clannish nature and ruthlessness of the Italians soon led them to take over the majority of organized crime, so much so that the few remaining groups of non-Italian gangs were soon forced to either align with the Italians or have their rackets taken from them.

Another big reason for the strength of the Italian groups in the US is the members frequently brought in their sons into the Mafia to strengthen group loyalty, whereas it was a lot less common in Jewish groups to bring in family members. This is still very prominent in the current mafia groups in Sicily, Naples (known as The Camorra), and Calabria (known as the 'Ndràngheta). You'll see lots of intermarrying between mafia groups: Mafia sons marrying mafia daughters. Examples in the US: Joe Bonanno's son Bill was married to Thomas Lucchese's daughter. In fact, the Sicilians are notorious for first cousins marrying each other, Carlo Gambino's wife Catherine Castellano Gambino (Paul Castellano's sister) was reportedly his first cousin. All of these things strengthen the blood ties of the criminal family and increase loyalty.

As far as French, German, or Scandinavian crime groups:

This depends on how you define "organized crime." It's possible there's evidence of groups from these ethnic backgrounds committed crimes in unison, though whether they'd be considered organized crime depends on how loosely you'd define that term. There was evidence of white, anglo-saxons committing bootlegging during prohibition, complete with bribery of police officers and politicians, but there's not much else that I've found so far [source: Ken Burns' Prohibition].

You're more likely to find evidence of non-Italians who were associates of the Italian Mafia in the US, as opposed to running their own gangs. Examples of this include: Frank Schweihs, aka "The German", in the Chicago Outfit or Joseph Watts with the Gambino Family in New York. Others outside the US include many French-Canadians in Montreal, Raynald Desjardins (French-Canadian) and his lieutenant Juan Fernandez (a Spaniard), being the most notorious. Both of these men worked with the Rizzuto Family, an Italian group in Montreal.

Please let me know if you'd like any more context or if you have any questions.

[More sources: "The First Family" by Mike Dash, "Cosa Nostra" by John Dickie, "Five Families" by Selwyn Raab, "The Sixth Family" by Adrian Humphreys, and "Mafia Inc." by Andre Cedilot]

Can I expand the original question to ask about Polish syndicates?

I know there were Poles and Polish-Americans who clashed with Al Capone (like Hymie Weiss), but I don't know if there was any kind of organization, or if they just happened to grow up in the same neighborhoods as other gangsters.

There was the French connection, which sold heroin on the US market. Heroin was smuggled from Turkey to the US through the French port Marseille, where they operated. I don't know if it's a satisfactory answer as the leaders themselves operated from France (they were from Corsica by the way) but that is the only recognized French crime organization known to have operated in the US.

Swedish Fredrik Lundin built a crime syndicate in Chicago that was taken over later by Al Capone in the 1920s and was also a republican congressman.

Gustav Skogens (2011). "Svensk styrde Chicago före Al Capone". Allt om historia (1/2011): p.46–49

I'm unsure but I think that one of the rival gangs to the Westies in Boston was a German syndicate