I'm... assuming you want to know the proper pronunciation. If you just want the closest pronunciation, it's "groo-yair". If you want more details, read on!
I am French. I have never, ever heard any American being able to properly pronounce the french vowel "u". It seems to be just as hard for Americans to pronounce the French "u" as for French people to pronounce the American "r". It's pretty much just as hard for Americans to pronounce the French "r", so with a word like "Gruyère" you're really asking for trouble! :lol: To give you an idea of what you may sound like, next time you meet a French person, ask them to say "Error". You should get a good laugh.
To describe the vowel "u" is a difficult thing to do. I think it's in there somewhere in the American word "few". You pronounce an "f", then you slowly morph "ee" into "oo". About halfway between that "ee" and that "oo" there's almost a "u". You really have to purse your lips way forward into a tiny circle. Almost like you're trying to whistle. It's a typical French sound where all the sound is forward in the mouth, and the cheeks and the lips are really tense. Since most of the American language is pronounced with a relaxed jaw, cheeks and lips and placed way in the back of the throat, it's not a sound you're used to make.
Now the French "r" I really can't describe. It's not rolling, but it does need some of the same kind of contact between the back of the tongue and the palate. Real hard for any American.
Everything else should be pretty straight forward:
G as in Grand
y as in Yeah
è as in met (as in we met in front of the statue...)
Here's a video where the word is pronounced correctly twice at around 1:21mn:YouTube - Beaufort : Prince des Gruyères