How do you pronounce 'fianchetto'?

Yeah, but if you pronounce it correctly, the waitress will look at you funny nine times out of ten.  You may even get "corrected."  ...At which point you should definitely order their best "chee-aunty." ;-)

fee-yan-ke-tow

Sorry, Chucksters - It's not "fee-ann-ket'-toe", it's "fyan-ket'-toe".  (You're welcome!!)  At least georgez consulted the International Phonetic Alphabet and got it spot-on!!  (There IS intelligent life on Earth!)  Thank you, georgez!!

Fee - En - Chet - Doe

Like Giorgez says, Italians do NOT say "fee-an-ket-toe" with a long ee in the start.

They instead pronounce it with a "ya",  a true diphthong. 

The link below takes you to recordings of the word as spoken by four natives:

https://it.forvo.com/word/fianchetto

Is that right? I think I once saw it claimed to be something like peertz.

Correct

Don't trust Dictionary.com then.  In Italian the double t makes for the t being held for a split second, the start of the t is barely heard. It somewhat like french and then the t starts again with a very definite t sound like one would start the word toe.  You could also say it is like the cockney pronunciation of the word kettle except the 2nd t is distinct t sound (again like the word toe) If the ch isn't an English k sound you are allowing the word to be colloquialized (turned into slang is what I mean).

Different languages and dialects pronounce double t's differently but if you change the Italian ch into an English ch why not also turn it into a Portuguese ch which sounds like an English sh?

This might depend on which dialetto italiano one is speaking and how fast and what emphasis the speaker wishes to convey.