Futurama had Native Martians (a clear analogue of Native Americans) talk like this. Amy's dad, Leo, spoke in the same way, leading to a bit of a Lampshade Hanging when he read a note from the Martians.
Leo Wong: Me know it them, 'cause they no use good grammar.
Also in Futurama, there are the Amazonians, a race of giant cavewomen. Also, when the robots rebel against the humans in one episode and all technology is lost, Leela reverts to speaking like this. "Leela bring fire?" "No, we're set for fire, thanks."
Then for one episode, everybody spoke like this when the brainspawn drain the intelligence of everybody on Earth except Fry.
And then there's Fry, who - being Fry - just does this randomly.
Bender need brain for smart-making!
The Lord of the Rings spoof Edward the Less features a barbarian strongman who speaks in the typical caveman dialect. However, he explains that he can use articles and personal pronouns, and just finds that they take up too much time in an eloquent speech that is nonetheless completely lacking in articles and personal pronouns.
Skwisgaar Skwigelf and Toki Wartooth from Metalocalypse, being Scandinavian, often add 's'es to words that don't need them (or leaves them offs of words that shoulds have them).
"Let me explains again, in perfectly clear Englishs, I wants flies in on a dragons, okay? How many times I got to tells this peoples?"
This, interestingly enough, is perhaps best explained as overcompensating for a difference between the Scandinavian language(s) and English - plural is not indicated by s in Scandinavian.
Aqua Teen Hunger Force does this with Oog the Caveman. When Frylock's supercomputer spontaneously travels back in time, Oog finds it, and...:
"So there me was beating boulder into powder because me couldn't eat it, and magic ball land in lap. Naturally me think, "All right, free egg." because me stupid and me caveman. So me spent about three days humping and bust open with thigh bone so me could eat it good, then magic ball shoot Oog with beam, and next thing me know, me go out and invent wheel out of dinosaur brain! Magic dino wheel roll for three short distance until me eat it; the point is, me get smarter. Soon me walk upright, me feather back dirty matted hair into wings for style, and me stop to use bathroom as opposed to me just doing it as me walk."
Oog simultaneously subverts this trope by speaking in a very clear, precise manner (save for the occasional outburst).
Incidentally, he claims "master english language" on his list of accomplishments.
See the example under Comic Books, but in the cartoon Teen Titans adaptation, Starfire explains/retcons: her race learns languages through "lip contact," selling the kisses as platonic, or even less. Hilarity Whiplash ensues.
The Transformers: Me Grimlock thinks me should be here too! (The other Dinobots speak the same way. In the comic, though, only Grimlock talks this way. Comic Grimlock, however, is not stupid. Depending on the Writer, it's Obfuscating Stupidity or a broken speech processor Grimmy never bothered to fix.)
On a related note with Beast Machines — TANKOR SLAG BEAST BOTS! Subverted when Rhinox's spark reawakens, giving Tankor a signifigant IQ boost — enough IQ to start plotting the downfall of Megatron and Optimus Primal. To hide what's happened, he still talks like he did before to most people, good guys and bad.
Strika and Obsidian do as well when also pretending to be idiots instead of the master tacticians they actually are.
The clones of the Gargoyles Broadway, Brooklyn, Lexington and Hudson speak this way. Lampshaded when Talon offers to teach them how to use free will "and verbs".
"Me so hungee" and "Me fail English? That's unpossible!" from The Simpsons.
In another episode, where Lisa grows an entire civilization in a dish with a tooth and some soda in it, a spokesman says, "We learned to immitoot you exarctly."
Subverted for laughs in an episode of Harvey Birdman, Attorney at Law, where the main cast travels back in time due to a freak accident. They expect this to happen when they are captured by cavemen. Instead, the cavemen speak American English and have academic degrees.
Wulf from Danny Phantom speaks in very broken Esperanto, though it's unclear if this is because he's an anthropomorphic wolf ghost (ghost wolf?) or simply due to sloppy translation.
The Piplings in the UK/Canada kids' show Waybuloo. (NB since it's a mixture of CGI and live action I'm not sure whether Waybuloo should go here or under Live Action TV, but the Piplings are CGI characters, so...)
The old Rocky and Bullwinkle spoofed this. In a Peabody and Sherman short, an evil man is trying to kill them, so he enlists some natives to do the job. He speaks to them in broken English, making incredible racist assumptions. "You make tiny buffalo and brave go dead. Arrows and tomahawks. Die die." Cue confused looks from the natives, until one figures out what he means, and explains the message to the rest of the tribe with a New Yawk accent. "He wants us ta whack da dog and da kid."
Parodied in Scooby-Doo! Mystery Incorporated, with the Mighty Whitey pastiche Rick Spartan and his sidekick Cachinga. Spartan insists on speaking to Cachinga in pidgin English and generally treating him like an ignorant savage, despite the fact Cachinga is a British Oxford graduate, real name Charles Wheatlesby.
I.R. Baboon from I Am Weasel talks like this to illustrate his idiocy. In "A Troo Storee", other people apparently start speaking the same way when they read I.R.'s book.