It's possible that it's All There in the Manual, but you won't get any hints in the show itself. Their name doesn't give any hints, it just Sounds Foreign, and their accent doesn't sound like that of any real country.
They're likely to talk about The Old Country a lot without ever telling you what it is. On the other hand, they have an increased chance of Laser-Guided Amnesia: you can't disclose where you came from if you can't remember where you came from.
Can be played for drama to hint at a character having a Mysterious Past, but more common as a way to create the ultimate Funny Foreigner - you can use all the best stereotypes without offending anyone!
Other characters may have their own assumptions, but these will be met by conflicting evidence, disagreement from others and offended denials by the foreigner himself ("Hungarian!? The Hungarians are our mortal enemies!"). The author, if pressed, will probably respond with a Shrug of God.
A popular variant in the UK is the "Eurotrash" character, someone whose origin can be loosely classed as Central/Western Europe but no narrower than that, speaking fluent English with an accent that hopscotches between Swedish, French, German, Italian...
Compare Ambiguously Brown, which can overlap with this trope. Contrast But Not Too Foreign. See also Where the Hell Is Springfield?, for when it's the setting of the show that's ambiguously located. The real life version as applied to versatile actors is Plays Great Ethnics.
- Lampshaded with the GEICO Gecko, a tropical lizard with a Cockney accent - one ad centers on people asking where he's from, and it deliberately cuts off before he answers.
- Wasn't he originally voiced by David Hyde Pierce in a very twee Northeastern American accent?
- Kaere/Kaede Kimura from Sayonara, Zetsubou-sensei went on an exchange program overseas and then back to Japan. To which country is anyone's guess, at first it seemed it was to America, until she started talking about some... unusual customs in whichever country it was.
- Kaolla Su from Love Hina is a perfect example, having her place of origin a secret until the second-last book in the manga.
- Yotsuba Koiwai from Yotsuba&! She has the amnesia, but with the excuse that she would have been a toddler when she left.
- The eponymous character of Yotsubato. She's known to be an adopted foreigner from "an island to the left".
- Serge in the Beverly Hills Cop franchise. What the heck was that accent supposed to be?
- Janosz Poha in Ghostbusters II spoke with a non-specific (Eastern European?) foreign accent but claimed to be from the upper west side of Manhattan. Several characters remarked on the absurdity of this claim.
- Nazo in Big Daddy. All we ever learn about him is that he's illegal, semi-literate, and is fond of "lamb and tuna fish" (yes, together). He's played by the half-Jewish, half-Filipino Rob Schneider.
- The Warrior's Way: Yang's country of origin is never clear and seems to be a general mish-mash of Asian cultures.
- Ben Jabituya from Short Circuit. He has a generic "Hindu"/"Indian"-sounding accent, is Ambiguously Brown, and can safely be assumed to be an immigrant, but not even his long-time friend and coworker Newton Crosby knows his origins. Ben does little to clarify things when directly asked:
Newton: Where are you from, anyway?
Ben: Bakersfield, originally.
Newton: No, I mean your ancestors.
Ben: Oh, them. Pittsburgh.
- In Game of Thrones, Tyrion asks what kind of accent Shae has, and she simply replies, "Foreign." Cersei suspects she's from the Free City of Lorath, but she doesn't directly confirm or deny this.
- Ax in Animorphs comes off as this in-universe in his human morph. As it contains DNA from four different people (two of them white, one Latino and one African-American), and he introduced himself as being from various countries, culminating in Canada ("I am from Canada. I am Canadese.")
- Errand from The Belgariad serves as a High Fantasy version of this. Many of the older characters also apply, such as Belgarath (who forgot what race of people he was).
- Queequeg from Moby-Dick, hails from a fiction South Seas island called "Kokovoko".
- The "Channel 9" broadcasters on The Fast Show were vaguely Eastern European/Mediterranean, but their language was a mixture of pure gibberish and British cultural references. Wherever country they were from, all we know is that the weather is consistently "scorchio".
- Eurotrash was full of this.
- Fez from That '70s Show is from some ambiguous foreign country (probably from somewhere in South or Central America). His real name isn't even available as a clue to his origins.
- In case you're wondering, his nickname is actually a (quasi-)acronym, short for "Foreign Exchange Student." But since it's not written in capital letters, many viewers probably assumed he was Turkish.
- Ishboo was a recurring foreign exchange student on All That with an abundance of bizarre cultural traditions that he'd insist others join in. Whenever someone asked where he was from, he'd brush off the question. It was sometimes implied that he wasn't a foreigner, and was just messing with everyone.
- Dr. Nick Riviera in The Simpsons. He also has a case of What the Hell Is That Accent?.
- Rolf in Ed, Edd n Eddy. It's implied, though never outright stated, that he's from somewhere in Europe.
- Family Guy made a Cutaway Gag about two guys who don't really have much of an accent, but speak in a way that you can tell that they're not from the US. They've become occasional Recurring Characters.