One of the more notable American detective novelists was a fella named Dashiell Hammett. He wrote private detective stories which mostly took place in San Francisco and sometimes New York. He was early to the game. He wrote mostly in the late 1920's. Characters from his most popular novels (The Thin Man and The Maltese Falcon) gained mainstream notoriety when they were made into movies in the decades following. Humphrey Bogart actually played Sam Spade in the Maltese Falcon.
In addition to his written work, he was quite a character himself, a veteran of both world wars, a tuberculosis survivor, a man once blacklisted as a communist, and had spent time in prison.
If you'll kindly forgive the Wikipedia reference at the bottom of this post I humbly submit Edgar Allan Poe's detective C. Auguste Dupin for your consideration. Someone else is much more qualified to comment on the accuracy of the Wikipedia article's claims about his overall influence on the genre but there is no doubt that he's a detective character and invented by an American author. Sadly, we are far from my area of expertise so that's all I can offer you! I'm just a Poe fan. http://en.wikipedia.org/wiki/C._Auguste_Dupin
Commissaire Jules Maigret is a Parisian detective who appears in 75 novels and dozens of short stories. He smokes a pipe and wears an overcoat, but does not have a deer hunter cap.
There are the novels about Nero Wolfe, written by an American named Rex Stout.
A personal favorite of mine is the novel "Celebrate Cases of Judge Dee" a semi-historical 18th century Chinese novel. Here's the wiki page on it:http://en.wikipedia.org/wiki/Celebrated_Cases_of_Judge_Dee
There are several interesting things about it. It is set in the Tang dynsasty hudnreds of years before it was written and is filled with elements from the then present Ming dynasty, it' s part of a larger genre of song'an Chinese mystery stories about judges who solve crimes, the structure is quite different then modern mystery novels (the murder is shown at the beginning, it's all about how Judge Dee will find the perp, a full confession under torture is necessary for conviction), but is also a pretty good read.
There's a sequel series of sorts by the English translator written in the 20th century, who tried to stick to the original style. They're pretty good too.
The detective novel as we know it - even if it started with Holmes in 1880s Britain - began in the urban mystery books of the earlier 19th century. Authors like Dickens, Balzac, Eugene Sue all depicted urban Europe (Paris and London particularly) as places whose depths could never be properly sounded. Many authors like those I mentioned were interested in the variety of life that co-existed in urban conglomerations - which, of course, were on the rise in the mid 19th century. Paris doubled its size in a mere 40 years. The uneasiness that came with living in a city where you didn't know everyone, coupled with the rising crime and vagrancy rates meant that urban fiction represented the city as a scary, unknowable place.
If you're looking for something more like a detective novel from another culture, I would recommend Emile Gaboriau's "Monsieur Lecoq." I highly doubt you'd be able to find a translation, but you could certainly try! The novel was written in 1869, so a decade or two before Doyle was writing. The first part of the book shows the reader a crime scene and introduces us to Lecoq, a young but perceptive police inspector. He does all the things we now expect a detective to do, like throw out complicated theories based on minimal observation. He quickly discovers that the prime suspect is none other than a peer of France, a Duke living in a mansion on the other end of town from a murder crime scene.
The second half of the book is a sort of "prequel," that describes the early life of the Duke and, through simple novelistic narrative, eventually explains who was at the crime scene and how they got there (the second part literally begins 30 years before the crime and slowly moves up to present day).
While the first part of the book leads you to condemn the suspect, the second part makes you sympathize with him, and realize that crime is more complicated than just logic and deduction. In the end, the inspector decides to let the man go after having fully understood his motives.
It's a great instance of early "detective" novels (or I'd call them more aptly police novels, since the idea of the gumshoe hadn't really developed yet!) since you can see the slow move from urban bildungsroman to crime mystery.
The "Philip Marlowe" books by Raymond Chandler are fantastic. The Long Goodbye is my personal favourite but they're all great.
Commissaire Jules Maigret by Belgian writer Georges Simenon is very famous in the francophone world.
I am not a historian, but you might be interested in Edgar Allan Poe's detective Dupin
How about Henning Mankel's Kurt Wallender books and the TV series in Swedish or the English version. Or the Italian Montalbano series perhaps?
Film director Satyajit Ray created Feluda and Shardindu Bandopadhyay created Byomkesh Bakshi - both Bengali (India) private eyes.
I quite like Tony Hillerman's Joe Leaphorn & Jim Chee series - both are officers in the Navajo Tribal Police, so the books feature a lot of native American culture/practices.
There's a piece of Biblical apocrypha called Bel and the dragon, an addition to the book of Daniel, which is arguably an early example of a mystery story. It's divided into two stories, that of Bel and that of the Dragon (convenient naming), but I'll only talk about Bel here.
In it, Daniel is hanging out with king Cyrus the Persian, who is said to worship a Babylonian idol called Bel. Cyrus is quite impressed with Bel: every night a huge offering of food and drink is placed in a sealed chamber of some description and every morning, lo and behold, the provisions are gone.
In spite of this, Daniel persists in his worship of the Hebrew God, so Cyrus proposes a deal. If the priests of Bel can show that it is the idol who is eating the offering, Daniel is to be executed. If Daniel is capable of showing that someone else is eating the offering, it is the priests who die.
The priests aren't particularly worried, and even allow Cyrus himself to lay out the offering that night, and seal the door with his own signet. Cyrus lays out the offering, but before the door is sealed, Daniel cleverly dumps a whole bunch of ashes on the temple floor.
Next morning, Daniel and the king discover that not only is the seal unbroken, but the food is gone. The king is quite content with this, but Daniel will have none of it. He discovers some footprints in the ashes and shows them to the king, who has the priests arrested. They confess to sneaking into the room through a secret passage, and are promptly executed.
I think my summary may have been longer than the text itself...
Anyway, as far as detective stories go, this one isn't particularly exiting: the big reveal is given out by the narrator before the protagonist discovers it, and the suspense of discovering the food gone is kind of undermined by Daniel's calm demeanor throughout the whole episode. It's still kind of interesting in that it reads like an ancient, religiously inspired, Sherlock Holmes story.
That said, someone with better knowledge of Biblical literature might have something smarter to say about the text's merits, or indeed its origins. The most intelligent thing I can say about that is that the only full extant versions of the text seem to be in Greek, and Aramaic versions appear to differ from those.
Crime and Punishment by Dostoevsky was one of the first detective novels.
Various radio shows of the 30s, 40s, and 50s (some of which were also books, comics, TV shows, movies), like:
These can be heard for free (but with subscription, except for the current daily show) here:
https://www.radiospirits.com/onradio/wrw_past.asp
And The Adventures of Harry Nile, which is still in production:
http://jimfrenchproductions.com/zc137m/index.php?main_page=index&cPath=1
In China, there is a traditional literary genre called Gong'an (公案小说), sometimes translated as "crime case," which shares many similarities with the modern detective genre. Both genres involve a protagonist that solves crimes. The key difference is that the protagonist of a gong'an novel is a government magistrate, whereas the protagonists of modern detective novels are (usually) private detectives.
The gong'an genre dates back to the Mongolian Yuan dynasty, when stories of Judge Bao (Bao Zheng 包拯) first circulated. The best known gong'an stories in the West are the Circle of Chalk, which was famously adapted by Bertolt Brecht as the Caucasian Chalk Circle, and Robert Van Gulik's translations of the Judge Dee stories. The recent Hong Kong film Detective Dee and the Mystery of the Phantom Flame in 2010 is a more contemporary adaptation of the Judge Dee novel.
If we're allowing for modern characters, one that I have not seen mentioned is my personal favorite; Jeffery Deaver's character Lincoln Rhyme. Both the author and character are from the US, and the character has been portrayed by Denzel Washington in the film The Bone Collector.