I'm just starting my PhD, looking at the prosecution of Axis war criminals in the wake of the Second World War. From my initial research, there isn't actually a large amount of literature on war crimes prosecutions outside of the International Trials at Nuremberg, Tokyo, and the further Nuremberg trials conducted by the United States. There is a growing body which examines the prosecution of Japanese war criminals, especially by Australian historians, and in Hong Kong. Have a look at the sources listed for some excellent examples of these.

One of the biggest contributing factors in deciding who was prosecuted at the end of the war was post-war aims of the prosecuting powers (the Allies). In 1942, the formative UN released a statement to the Axis leaders that they would be tried for their complicity in initiating aggressive wars, and that military personnel would be tried for the commission of war crimes. Between 1945-1948, over 60,000 individuals were prosecuted, in both national trials, conducted by individual nations for crimes committed against their peoples, and more generally by International Military Tribunals, for crimes against peace, crimes against humanity, and general war crimes. Choosing who to prosecute, and the sentences imposed on those found guilty, was influenced not only by available evidence, but also the post-war aims of the prosecuting powers. As such, the Japanese Emperor was spared prosecution, to ensure that Japan retained a central figurehead for the populace to gather round, ensuring a relatively successful occupation by the Western Allies. Similarly, the members of Unit 731, the notorious Japanese medical unit that conducted widespread atrocities against the Chinese, were given safe passage to the Untied States, to share the results of their gruesome 'experiments'.

The German Field Marshall Albert von Kesselring was tried for failure to maintain discipline amongst his subordinates in Italy, who had massacred civilians in retribution for partisan attacks. He was found guilty, and sentenced to death. However, the desire amongst the Western Allies to rearm Germany, and to minimise indicting the entirety of the Wehrmacht by proxy, meant that this was commuted to a prison sentence. This choice of commutation has in part been linked to the formation of the "Clean Wehrmacht" myth.

As we can see from the above examples, the initial pursuit of justice was influenced by external political decisions of the Allies. As for why Italian atrocities seem to be less well known that their German and Japanese counterparts, the fact that Italy changed sides during the war cannot be ignored.

(EDIT: As u/AlviseFalier has noted, this is a gross oversimplification. I'm sure that they could provide a much more nuanced and accurate description of what happened).

As a technically victorious power, they were not privy to the same level of scrutiny as the clearly defeated German and Japanese militaries and political establishment. There was also actions taken by the Italian government to move past the crimes committed by the Fascist government, in an effort to unify the country in the immediate post-war years. A great number of public servants and politicians were likely complicit to some extent in war crimes. However, the British were concerned of the popularity of the Italian Communist Party, needed these people to maintain a strong and stable government to counter the communists. The British, having a presence in post-war Italy (the British military tried Kesselring in Italy), ensured that requests by Yugoslavia and Greece to extradite alleged war criminals were denied, or ignored. The failure of the British and Italians to prosecute potential Italian war criminals has lead to the belief amongst some that Italy was essentially a victim of the Germans, leading to further repression of the memory of Italian atrocities.

Sources.

Sandra Wilson, Robert Cribb, Beatrice Trefalt, and Dean Aszkielowicz, 'Japanese War Criminals: The Pursuit of Justice After the Second World War", Columbia University, 2016.

Kirsten von Lingen, "Kesselring's Last Battle: War Crimes Trials and Cold War Politics, 1945-196)", University of Kansas, 2009.

Yuma Totani, "Justice in Asia and the Pacific Region, 1945-1952: Allied War Crimes Prosecutions", Cambridge University Press, 2015.

Suzannah Linton (ed.), "Hong Kong's War Crimes Trials", Oxford University Press, 2013.

Effie Pedaliu, "Britain and the "Hand Over" of Italian War Criminals to Yugoslavia, 1945-1948", Journal of Contemporary History (2004), Vol. 39, No.4.

Paul Taucher, "Command Responsibility at the Sandakan-Ranau War Crimes Trials", Murdoch University Honours Thesis, 2016.