I teach English to kids in Korea. When they make fun of my speech, they throw in lots if “urs” and “uhs.” Lots of flat vowel sounds.
‘Look at me, everybody! I’m a cowboy! Howdy, howdy, howdy!’
I’ve always loved that one.
" Rather rather rather."
Oh yeah, that reminds me of another one. They overemphasize our lax vowels over the tense ones that seem to be present in a lot of languages (granted I’ve seen certain flavors of Brits make fun of us for that one too). (Which, upon rereading, is what I think you were saying… oops)
The “urs” are one kind of “r-coloring,” which is one of the things about English sounds that stand out compared to other languages. It’s when the formation of the tongue causes a retroflex articulation of a preceding vowel sound. “Coat” becomes “court,” “shit” becomes “shirt,” and “cat” becomes “cart.”
The “flat” vowels are called “lax” vowels. Your mouth is comparatively relaxed when you produce them, as opposed to “tense” vowels, for which you need to tighten your mouth.
“Bit” vs. “beet.”
“Cot” vs. “cat,” etc.
Many languages (Spanish, Japanese, Korean) don’t really have r-coloring or lax vowels, so that’s what really stands out about English for people who speak those languages. English sounds “lazy” to them.
How you pronounce your vowels in English – especially at the ends of words – often depends on what neighborhood you live in. It’s almost the same with a final “r.”
When I was in Denmark, I had trouble with a “d” sound. I know that it was more like a “th,” but is it the “th” sound as in the English thin or then? Whatever it was, I never got it right. My hosts just shook their heads sadly when I went to their shop in Odense.
Something like around 0:35 in this clip…
You hold a pen beween your teeth (as if it was a cigar or a pipe) while speaking.
I realize the OP is asking about non-Americans, but just for fun I tried to see if I (as an American and native speaker of English) could construct some English-sounding but meaningless sentences.
I tried to mimick typical word lengths and letter combinations and such, but it’s really hard to sound natural while avoiding using too many actual short English words.
Here’s the best I could do:
A bon do pow da bade felmish, ta bits it moe nast aslempt. Ips mock the grabe, thoe. Ole ket, a mest bo ta ray tor.
The best fake English video I know is this video. (starts at 1:00)
Americans trying to speak foreign do some strange things with Rs, though.
You don’t say “una cerveza, por favor” as “oona serbesa por fabor” or as “oona zerbeza por fabor,” you say it as “oooonaah sseRRRRRbesaah poRR faFoRRR.” All those are soft r’s in Spanish, but you guys sound like you’re trying to imitate cars raring up before the start flag comes down in an F1 race. I’ve heard Americans do similar things in French and German, but my French and German aren’t good enough to write what they were doing.
I don’t think I’ve heard someone in Spain trying to fake an American speaking English in ages, usually it’s an American (or any other foreigner as befitting the story) speaking Spanish. The “American speaking Spanish” version treats each syllable as a separate word, splits them in the wrong place, stresses the wrong ones, can’t pronounce a z or j to save his life, speaks very loudly and does things like say “siuda!da!no! america!no!” (american citizen!) waving his passport when he can’t get his way (something which, by the way, I have encountered in reality in Barajas).
On an only slightly related note - I was in England this past week and one of the women who served me breakfast thought I was from Canada - this surprised me, because I wouldn’t have guessed that I sound Canadian at all. (Although I’m not sure I would know what it means to sound Canadian.)
Khadaji, perhaps it had to do more with your general demeanor and her contrasting experience with the attitudes of American tourists versus Canadian.
Add -ation to everything. Quiero una cerveza would be “Queration una cervezation” in Mexican Fake English.
Yeah maybe. She said that Canadians have a “typical” sound that I also had, but she may not have wanted to say the real reasons - which may have been less than flattering to Americans.
And (some) Canadians can get quite tetchy when mistaken for Americans, so if one is unsure it may be better to err on the Northern side. 
She was from Poland working in the UK. It may simply be that all North Americans sound alike to her. I would likely have difficulties pinning down which Slavic accent I was hearing.
I’ve heard that Koreans say “shella shella shella” to imitate English speakers, in the same way some English speakers use various combinations of “durka”, “Muhammad” and “jihad” to imitate Arabic.
I’m thinking English gibberish as spoken by a non-english speaker might sound much like Frisian.