When in Rome, salute as the Romans do:

The Roman salute is the oldest known hand salute. It consists of holding the right arm straight out from the shoulder, elevated about forty-five degrees. It was widely used throughout the world until World War II. In the United States, civilians gave the Bellamy salute, based on the Roman gesture, while reciting Francis Bellamy's Pledge of Allegiance. It wasn't until 1942 that the United States Congress abolished the extended-arm salute in favor of the current hand-over-the-heart gesture. That decision was in reaction to events in Europe; Benito Mussolini and the Fascist Party of Italy, seeking to revive the spirit of the Roman Empire, had adopted the Roman salute in the early 1920s. Adolf Hitler copied it, and it developed such a close association with Nazis that it has never been used by any organization not specifically linking itself to the Nazis since then.

Interesting that it took American legislators over 20 years to see the hand, writing on the wall.

It's not quite extinct either:

...mostly because of tradition, we've been using it since the early 1930's when the Communist used the closed fist salute, the opposite of that being the Roman Salute .

The American Falangist Party discourages Party members from using this salute too much in public until people become more informed.

On the other hand (no pun intended), this gesture may have been a case of life imitating art:

The raised-arm salute is one of the best-known symbols of Fascism, supposedly based on a classical Roman custom. But no Roman work of art displays this salute, nor does any Roman text describe it.

Well before Fascism, the [Roman] salute frequently occurs in films set in antiquity, such as the American Ben-Hur (1907) or the Italian Nerone (1908), although such films do not yet standardize it or make it exclusively Roman. In Spartaco (1914), even Spartacus uses it. In imitation of such historical films, self-styled “Consul” Gabriele D’Annunzio appropriated the salute in its now familiar form as a propaganda tool for his political aspirations upon his occupation of Fiume in 1919.

From a review of the book D'Annunzio - The First Duce:

Virtually the entire ritual of Fascist politics made familiar by Mussolini--the balcony address, the Roman salute, the dramatic dialogues with the crowd, the use of religious symbols in a new secular setting--was influenced by D'Annunzio at Fiume. Both were masters of a political style based on personal charisma. Each spoke for a "new" Italy and, eventually, for a new world. Each attempted to transform his countrymen into more heroic types by an ethic of violence and grandeur.

More about D'Annunzio.

Actually, I have found that the Roman salute does occur in several works of art. Just go to Google, and search for images with "Roman salute" and you will find a few.

Was the National Socialist German Workers' Party influenced by the U.S.A.? In 1892 a self-proclaimed socialist created the pledge of allegiance and the original salute to the U.S. flag. The salute included an outstretched arm (the only place with secret photos of the original socialist salute to the U.S. flag is http://rexcurry.net ). The National Socialist German Workers' Party was aware of the salute to the U.S. flag when it adopted its salute.
Government schools in the U.S. followed a policy that was later adopted by the National Socialist German Workers' Party: punishing and expelling children from government schools if they refused to perform the straight-arm socialist salute. In 1940, in Minersville School Board v. Gobitas, the Supreme Court ruled that a government school could expel children for refusing to perform the straight-arm socialist salute to the U.S. flag. The National Socialist German Workers’ Party had been in existence since 1920 (with electoral breakthroughs in 1930 and dictatorship in 1933, and WWII in 1939). About three years later (1943), in West Virginia Board of Education v. Barnette the Supreme Court reversed itself and decided that school children may not be forced to stand and salute the flag. http://members.ij.net/rex/pledge_lawyer.html
The U.S. salute was so similar that the U.S. changed its salute after the monstrous National Socialist German Workers' Party tried to impose socialism in Europe.
Francis Bellamy, the author of the pledge, was the first cousin of another socialist, Edward Bellamy. Edward Bellamy's futuristic novel, "Looking Backward," published in 1888, described a totalitarian Boston in the year 2000, that Edward Bellamy portrayed as a utopia. The book spawned a socialist movement in Boston known as "Nationalism," whose members wanted the federal government to nationalize most of the American economy. Francis Bellamy was a member of the "Nationalism" movement and a vice president of its socialist auxiliary group.
The book was translated into 20 foreign languages. It was popular among the elite in pre-revolutionary Russia, and was even read by Lenin's wife. John Dewey and the historian Charles Beard intended to praise the book by stating that it was matched in influence only by Das Kapital.
The Bellamys saw government schools as a means to their socialist "Nationalism." In the book "Looking Backward" government schools are the way that the government forces everyone into its "Industrial Army." Those were views later shared by the National Socialist German Workers' Party. For more info on the horrid National Socialist German Workers' Party see http://members.ij.net/rex/swastikamain.html
The self-proclaimed socialist Francis Bellamy proposed the original socialist salute and the pledge after he joined the staff of the magazine "Youth's Companion." The salute and pledge were created by Bellamy to promote socialism among the youth in the most socialistic institution -government schools. When the U.S. Constitution was written, most children received private educations. Bellamy lived during the time when schools were becoming socialized heavily in the United States. Bellamy believed that the best way to promote socialism was by removing children from their parents and placing them in socialized schools (government schools) and other youth groups. It was a view later shared by the National Socialist German Workers' Party and its youth programs.
Socialists use a sense of belonging to seduce people. Socialist schools (government schools) are used to destroy individuality and create dependence on government. Bellamy had often lectured on the so-called "virtues of socialism and the evils of capitalism." Adherents of the National Socialist German Workers' Party did the same.