Subverted in Babylon 5, where Zathras spoke in a primitive manner (actually based on creator JMS'sRussian Polish grandmother's way of speaking English, the syntax is unmistakably Slavic) but was, besides being capable with very advanced technology such as time travel devices, in a way, clear headed and philosophical, often seeing the big picture far better than most of the regular cast. This juxtaposition made for some classic lines, like:
"Zathras is used to being beast of burden to other people's needs. Very sad life... probably have very sad death. But at least there is symmetry."
"But only, Zathras have no one to talk to. No one manages poor Zathras, you see. So Zathras talks to dirt. Sometimes, talks to walls, or talks to ceilings. But dirt is closer. Dirt is used to everyone walking on it. Just like Zathras. But we have come to like it. It is our role. It is our destiny in the universe. So you see, sometimes dirt has insects in it. And Zathras likes insects. Not so good for conversation, but much protein for diet."
"Cannot run out of time. Is infinite time! You are finite, Zathras is finite, this... is wrong tool. No... ...no, never use this."
This also how Drazi speak. Ivanova hang lampshade in episode "The Geometry of Shadows":
"Just my luck. I get stuck with a race that speaks only in macros."
Star Trek: The Next Generation has the Pakleds, who speak in extremely basic sentences (though the grammar is correct). "We look for things. Things to make us go." (Their ship's navigation system has broken down. But not really — it's a trap for Geordi.) Although the Pakleds' ship is probably stolen, it's questionable whether they're as stupid as they sound — ever try to fly an airplane? Or to kidnap a military officer?
On the other hand, they are fooled by a very obvious ploy. So while they're not as dumb as they seem, they aren't much brighter than, say, TV executives.
The Pakleds are a textbook example of the importance of the Prime Directive. Clearly, some or another idiot species made First Contact with the Pakleds long before they were ready — and witness the result.
In a similar vein, an early Star Trek: Enterprise episode features a lost colony of humans whose language has "devolved" into a primitive form after 70 years of non-contact with Earth.
To be fair, everyone except for the very youngest children had died off all at once several generations back; so everyone living there now learned to speak from people who had barely learned to speak themselves, having no adults to teach them better.
There's the the epically cheesy "Brain and Brain, what is brain!" brought to us by the... questionable episode fittingly titled "Spock's Brain".
Or "The Omega Glory" where warfare reduced two nations to "tribes" speaking a mangled, devolved English.
Who could forget "Devil in the Dark", featuring the Horta, which at one point carves the words "NO KILL I" in the cavern floor using its searing-hot flesh?
Although that's more of a cast of a Starfish Alien trying to write in a language it just learned.
Overly literal translations of Klingon can come across this way, as the language lacks several things considered important to English grammar (like verb tenses).
And the verb "to be". Also, the language structure is backward from English.
Heghlu'meH QaQ jajvam. "die-someone-for be.good day-this" = "It is a good day to die."
Word of God is that English is something like a fifth language to him and he doesn't get the grammar.
Comically subverted in Jeeves and Wooster. Wooster blackens his face and poses as an African tribal chief using Hulk Speak. But suddenly the real African chief shows up (wearing the same costume!) and, as he is actually college educated in England, starts speaking in a perfectly normal, and somewhat high-class, English.
Heroes: Nerdy, childish Hiro speaks broken English with a Japanese accent. Badass, sword-wielding Future Hiro speaks perfect English with an American accent.
The first two movies of Lexx include Giggerota the Wicked, who has a very unusual manner of speaking, which sometimes follows this trope. In her case, however, it is unclear if she doesn't really know how to speak properly or if she's just affecting this as part of her outlandishly savage and feral persona.
Buffy the Vampire Slayer has a variation of this. Cordelia talks this way to Sven, the exchange student she's hosting in "Inca Mummy Girl". "Punch-y! Fruit drink-y!" But after she's out of earshot, Sven complains about it.
In "Bad Eggs" Buffy comes in exhausted after a night of making out with Angel...I mean patrolling for the notorious Gorch brothers.
Giles: How did the, um... hunt go last night, Buffy? Buffy: No go. Giles: Uh, 'no', 'no' you didn't go, or, or, or you were unsuccessful? Buffy: No Gorches. Xander: Apparently Buffy has decided the problem with the English language is all those pesky words. (looks at her) You... Angel... big... smoochies? Buffy: Shut... up.
Subverted in an episode of Jack of All Trades when Lewis and Clark, lost in the South Pacific, need help getting back to the Oregon Territory. Jack Styles hooks them up with an Indian guide, Sacagawea (yes, in the South Pacific), who orders them around in stereotypical Hollywood Indian. Once L&C are out of earshot, she tells Jack that dropping her articles and using other ungrammatical language increases her authority through intimidation.
Lucille's adopted Korean son in Arrested Development starts out saying nothing but "Annyong" (the Korean word for "hello"; the Bluths assume it's his name), but gradually learns English over the course of the series, talking in stilted phrases at first but with better grammar in later appearances.
Played with in an exchange between the three kids.
Annyong: He no have father? (about George Michael)
Maeby: No, he have father. Father no love him.
George Michael: Wait, he love me... loves me.
On Eli Stone, Eli's acupuncturist/guru Frank talks this way when he's in his "Dr. Chen" persona, which he adopts when dealing with customers who don't want to take their mystical wisdom from a guy with an American accent.