Taming of the Shrew, Two Gentleman of Verona, Merchant of Venice, Much Ado about Nothing, Titus Andronicus, Romeo and Juliet, Julius Caesar, Othello, Coriolanus, The Winter's Tale. Almost as many as were set in Great Britain. Was it simply the renaissance, or was there more to the popularity of Italian settings?
It's not the end of the story, because adapters have choices. But the bulk of the reason for the various Italian city-states' setting of Shakespeare's plays is most likely his sources. Good old Will took a lot of his source material from Italian literature, especially Boccaccio's Decameron. A lot of these texts are actually medieval, not "Renaissance," by basically every stretch of the imagination.
As with All's Well, Shakespeare is in general pretty faithful to the settings of his sources even when they are not Italy or England. (Hamlet perhaps being the most famous example.)
Hopefully the Roman Empire-related plays explain their setting rather inherently.
Added to the excellent previous answer must be added the fact that setting plays outside England allowed Shakespeare a greater degree of freedom in what he could say on stage.
Considering the way theatre was seen in the Elizabethan and Stuart era (to whit: very poorly; with the playhouses (especially after 1595) being located in the heart of the red light distract (in the case of The Rose, The Swan, and The Globe theatres) or in a part of London famed for being seen as run by criminals (Blackfriars- it gained its reputation after Henry VIII dissolved the Dominican friary who owned the region, selling the land to various landlords, but never updating the by-laws in the region; as such Blackfriars was under Church Law but with no church to enforce it; thus it became widely accepted that all warrants for any criminal could not be extended into Blackfriars; hence its reputation of criminality and its nickname to Londoners at the time, Alsace, after the contested region of France).
Given the low nature of the craft and its reputation for salacious, lewd and at times subversive writings, the average play-write had to balance between writing about power and running the risk of upsetting those IN power, or avoiding writing about anyone with power at all. Censorship was very real and arrest for causing dissent was equally real.
This caused writers to respond differently. Ben Johnson tended to avoid writing about power; masterpieces like Volpone and The Alchemist rather are set in contemporary London, and focus on the rude vulgar and canny street culture all around him, allowing him satire society without mentioning any named person.
Shakespeare liked to avoid setting his plays in London, but rather placed them in foreign lands, allowing him to comment upon society from a safer position.
Even then it didn’t always work. Johnson’s satirical tongue led to James I threatening to cut off his nose once and WS has to tread carefully after the gunpowder plot due to distant family comments ruins to the conspirators. Yet his ability to set a play in a foreign country allowed him (and others) discuss matters going on around him safely.
He used Macbeth (set in Scotland) to talk about Jesuit involvement in the gunpowder plot; he uses Ancient Greek tales of Troy to openly refer to the sex industry in Southwark (With the character of Panderus in Troilus and Cressida speaking directly to the prostitutes and pimps in the audience); and The Merchant of Venice allows him comment upon the trail and execution of the queens Portuguese Jewish Doctor, Lopez.
He responds to growing anger towards Germans living in the Steelyard with his famous insert into the script of Sir Thomas Moore (the brilliant condemnation of the Evil May Day Riots, which one cannot help but feel was influenced by him staying with a Huguenot family for many years in London), and we see time and again, he is able to avoid the censor by placing plays outside of London/England.
Foreign settings also allowed him change and include events to become far more salacious because they did not reflect London’s rather conservative if not outright Puritan manner (for example by setting Romeo and Juliet in Italy he could shock the audience with Juliet being 13 and her mother only 26, a fact mostly accepted because ‘its what foreigners did’ (gross simplification there but it does fit).
He wasn’t alone in this but it is worth comparing his and Johnson’s approach to censorship and always worth remembering the infamous performance of Richard II paid for by the Earl of Essex in 1601 just before his uprising. The Lord Chamberlains men sent the actor Augustine Phillips to appeal to the Queen that while the play did indeed include references to toppling a reigning monarch and while it was clear Essex had hoped the play would stir up Londoners to join him:
“He sayeth that on Friday last, or Thursday, Sir Charles Percy, Sir Jocelyn Percy, and the Lord Mounteagle, with some three more, spake to some of the players in the presence of this examinant to have the play of the deposing and killing of King Richard the Second to be played the Saturday next, promising to give them forty shillings more than their ordinary to play it.
Where this examinant and his friends were determined to have played some other play holding that play of King Richard to be so old and so long out of use as that they should have small or no company at it But at their request this examinatant and his friends were content to play it the Saturday and had their 40 shillings more than the ordinary for it and so played it accordingly.”
Shakespeare and co were believed, but crucially it shows the dangers of writing a play in a contemporary setting.
No, this was never the main reason for doing so (Elizabethan audiences were fairly familiar with classical tales of those from the Decameron) but it was a factor that no writer could ignore.
Sources; Wood’s In search of Shakespeare.