Indo-European Alien Language

Most alien (or non-human of any kind) languages in media are simplistic and based on the language of the creators of the media. Provided the languages have some form of grammar established, be it languages from fantasy creatures or aliens, they will always look more like English than even Welsh looks like English. It seems that even when aliens aren't speaking English, they're speaking something like it. In the Con Lang community, these alien languages would be described as a "relexification" of English, or relex for short -- many of these may count as fictionaries.

Some conlangs, however, go beyond that, and the author actually shows their work to some extent and creates a language with grammar that is different from that of English. Unfortunately, the result often still shows the typical features of Indo-European languages - similar inflection or conjugation patterns, similar use of copulae and auxiliary verbs, and so on.

As most writers are not linguists, this trope crops up unsurprisingly often across fiction. Of course, you would have to be extremely dedicated to create an entire language not based on your own at all -- and even if you did, only the particularly dedicated would try to learn it. Thus, it follows that most fictional languages look like English, particularly from the perspective of native speakers of Basque, Turkish or Hebrew, for instance. Not that that's necessarily a bad thing though.

What really counts is how an author uses their Con Lang in story. A story with a Con Lang that shares few similarities with an Indo-European language will still fail if it's a bad story. And remember that as much as authors can try to avert this trope, similarities to other languages is not a bad thing. And unless that particular Con Lang is going for The Unpronounceable, then all languages will share some very basic similarities.

If the author invents a language designed to avoid similarities to any real language, that falls under Starfish Language. Re Lex is a subtrope for when the invented language has what amounts to a one-to-one correspondence with the language the work is written in.

Film

 * Avatar: Averted by Na'vi, which has a tripartite alignment system (not very common at all!), five verb tenses, inclusive versus exclusive first person plural pronouns, ejective consonants, et multa cetera. Helps that the language was designed by a professional linguist who went to great lengths to make the language not resemble any one human language, while still being usable by human beings.
 * Mostly averted in Star Wars. People speak Basic (i.e. English) most of the time anyway, but Shyriiwook gets points for A: being just one language of the Wookiees, B: being very different from English when we hear it and C: being so hard to learn, Leia has to get help from a Wookiee with a speech impediment that makes the language easier for her. Indeed, in the movies it's based on bear noises. Other languages spoken such as Huttese are based on the sounds of languages like Aymara and Quechua (neither of which are Indo-European), although Huttese is not a conlang, just a bunch of random sounds. And, to top this off, there's a lot of bilingual conversations in the Star Wars movies and EU, with the fairly reasonable justification that some species' vocal apparatuses simply can't make the sounds of the other languages.
 * Bright Tree Ewokese is based on Tagalog.
 * According to Wookieepedia, the language of High Galactic was the original language of the Republic and Jedi Order before Basic. Some of the words, like "fi" for son, or "pera" for father, suggest that it is a Latin/Romance language expy. Which makes sense, since the Republic lives and breaths the Space Romans trope.
 * Mando'a, the Mandalorian Language, contains single words which hold several meanings, depending on the context in which they are used. For example, the word "Kandosii" can mean anything from "classy" to "awesome" to "dangerous" or anything else, usually taken positively. Mando'a is less about the words themselves and more about the emotion in which they are invoked.

Literature

 * Even JRR Tolkien fell into this despite his incredible dedication to his languages (he himself was a philologist). Quenya was originally based on Finnish grammar; however, Finnish is part of a language family called Uralic that is unrelated to Indo-European, but shares some similarities, and Tolkien negated any brownie points by removing all the parts of Finnish that made it so Uralic to start with. If we want to be didactic, the phonology was a mix of Finnish and Latin, morphology is totally Finnish, syntax is a mish-mash of Latin and Greek. Sindarin was based mostly on Welsh, a Indo-European language in the Celtic subfamily. Interestingly, he made a serious effort (and not too shabby of one either) to derive them from the same source language, even though this would be completely impossible for the languages they were based on.
 * Well, one of his early names for Quenya was "Elf-Latin". Tolkien knew what he was doing, and he apparently liked the Indo-European family (not to mention being more familiar with it). The result is that Quenya's structure is a lot like Latin, and Sindarin's is a lot like Spanish and/or Welsh.
 * Also the sound system was intentionally designed to be as euphonic as possible to a European language speaker (hence the overabundance of l, r, and n sounds and the relative lack of consonant clusters).
 * There is some reason for the Elvish languages to sound like Indo-European languages, since in-universe, all human languages are supposed to be derived from or influenced to some degree by the Elvish ones.
 * Then there's Khuzdul, the Dwarvish language. Tolkien never went to the trouble to construct more than a few words of Khuzdul, but it looks a little like Hebrew, which is not Indo-European.
 * Sometimes authors from non-Anglophonic countries embrace this trope, because due to influence of Tolkien and other Anglophonic writers, the audience is accustomed to the sound and flow of English as "proper" for a fantasy language. As an example, this is a reason (among others) why Polish fandom hates the Łoziński translation of Lord of the Rings

Live Action TV

 * Star Trek, at times when the Universal Translator still fails. All alien languages sound as if they obey English phonology, but their transliteration to the Latin Alphabet seems extremely implausible, using c and k interchangeably and the obligatory useless apostrophes.
 * Dominionese apparently has a passive voice transitive which serves as an optative mood. Obviously the writers just garbled some terminology together as this more or less implies you can't just politely ask a Vorta to die, you also have to specify the cause or person killing said Vorta. It might translate as "please die in a fire," or, "with all due respect, ma'am, you can go jump off a bridge".
 * Notably inverted in Klingon: though still a language a human can pronounce, it was made by a professional linguist and has both an alien grammar and unusual sounds. Unless, of course, you speak Tlingit, from which Marc Okrand borrowed precisely because it has sounds not in better-known/more widely spoken human languages.

Tabletop Games

 * The Ilithyrii, Darkelf or better known as Drow language is just basic English grammar (plurals end with n instead with s, female titles still end in-ess, etc) with new words based on a lot of hissing sounds (sibilants) like x and s to simulate their underdark home that's prone to echo "harder" consonants a lot.
 * Played with in World Tree RPG. The world's languages are based on a divinely created Common language with simplistic grammar, and all known languages are based on Common. So every language in that setting is like this trope towards every other, and described as being only about as different as English and Italian (ie. not very). We're also told that the standard pronoun "genders" are based on species, so that "the male Cani greeted the female Rassimel" would be written as something like "Ce greeted rir" instead of "He greeted her". The in-character journal by one of the game's authors has the hero dealing with other grammatical oddities like social-class markings, and even making fun of cheap in-universe novels that don't think through their "alien" language.

Video Games

 * The Tevinter language from Dragon Age is very similar to Latin, since Tevinter is based on the Roman Empire.