Googling says both “nitch” and “neesh” are correct. I used to pronounce it the former way for years. Now the former sounds wrong to me and the latter, while the correct way to pronounce the word in the language it was borrowed from, sounds too much like someone trying too hard, like pronouncing the “t” in “often”.
I’ve never heard it pronounced any other way but “neesh” and now that you’ve mentioned it, it does kind of sound pretentious and I’ll here on out be self conscious every time I say it. Thanks a lot, solost :mad: 
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Sheesh, it’s neesh! 
Consider the following routine from Abbott & Costello’s radio show, circa 1948 or so, which is contained on an Old Time Radio tape that I have owned since childhood.
Abbott: If you’re lucky, you’ll find your proper niche in life.
Costello: You’re right. Why, if I can find my-- What will I find?
Abbott: A niche, a niche. You’ll find a niche.
Costello: Abbott, when I find an itch, I scratch it!
Abbott: No, no, I’m talking about a niche in life. A niche is what everyone is looking for. Anyone who is successful has found a niche.
Costello: Well, if that’s the case, I know an Airedale who’s doing very well.
This routine makes no sense at all if “niche” is pronounced neesh. Therefore nitch is the correct pronunciation.
I say ‘nitch’ for the same reason I stopped pronouncing gyro “yee-ro.” Because to me “neesh” sounds a little pretentious. I do pronounce bruschetta “brusketta” though.
A little shelf built into the wall, where you put a statue, is a “nitch”.
The little place one carves out to give life meaning is a “neesh”.
Neesh.
Neesh, and there’s nothing pretentious about pronouncing French words correctly: at least not in Canada.
How do you pronounce quiche?
I follow the usage of Dr. Seuss.
That’s “nitches,” not “niches.”
“Kitch.”
Nitch, like bitch, if I am saying, for example, “She really found her niche.”
But in another context, like a tech company who isn’t leading the pack, then I will say, “You’re a neesh player.”
I have issues.
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Real men don’t pronounce “quiche.” 
neesh. (It’s the correct and ubiquitous UK pronounciation though, so probably not that helpful to you.)
Quickie, duh.
It’s neesh.
Though ‘Nietzsche’ has its charms.
To be honest, if I hear someone saying “nitch” I would be rather judgmental. Silently though, I’m not a complete dick.
See also: clique pronounced “click”.
Neesh or nitch are both fine. I think we should teach the controversy.
Reminds me of that poem often attributed to Dorothy Parker:
Upon my honor
I saw a Madonna
Standing in a niche
Above the door
Of the prominent whore
Of a prominent son of a bitch.
Dorothy Parker denied it! Hearst didn’t allow fornicating among unmarried visitors to his castle, and her beau tiptoeing to her room in the middle of the night was thwarted, according to one of David Niven’s anecdotes. This was supposed to be her getting back at him.
Hilarious, though.