According to testimony from Aladena Fratianno--aka "Jimmy the Weasel"--a Mafia boss turned FBI informant, the Chicago family of the Mafia ordered the assassination of the show's producer Desi Arnaz because they didn't like (a) the fact that the success of the show was focusing attention on the Mafia, and (b) the show's portrayal of Italians. Fratianno said that two hitmen hid themselves near Arnaz's house one night waiting for him to show up, but he never did. Shortly afterwards the assassination order was rescinded when it was realized that Arnaz's murder would cause the Mafia more trouble than it was worth.
To counter some of the claims that the show was anti-Italian, later episodes gave a more prominent role to Agent Enrico Rossi (the Italian-American member of Ness' team) in order to show an Italian-American as a hero working for law and order. Rossi was played by actor Nicholas Georgiade--who was Greek.
Nicholas Georgiade, who plays Untouchable team member Enrico Rossi, actually had a small, uncredited role as a thug in Westinghouse Desilu Playhouse: The Untouchables: Part 1 (1959), the series pilot. In the movie, after the Untouchables raid a plant and round up the criminals, Georgiade plays the thug identified as "Frank Cotter, a gunman from New York" who quips to Ness that he'll read about Ness' obituary in jail, thus getting punched by Ness.
In the early 1950s Desilu Productions was prohibited from using the word "pregnant" on network television. At the end of the decade, when this series was produced, although euphemisms were often used, the word "prostitution" was used numerous times in the series.
Among the concessions made to Italian-American groups, the network and producers were allowed to use Italians and Italian names for criminals based upon real persons but agreed that fictional criminal characters would be non-Italians.