No, what you're missing is the ability to roll your Rs. Don't worry, though, it's English's fault, not yours.
Fieri is of Italian descent -- "fieri" translates, roughly, as "proud" -- and Italian, like many of the other Latin-descended languages, features so-called "rolled" Rs. The result is a sort of purring sound, the result of the tongue rattling against the roof of the mouth as the letter is pronounced.
It's a sound that simply doesn't appear in the North American pronunciation of English (unlike, say, Scottish English, whose rolled Rs result in a lot of mockery -- see the Austin Powers films or a number of Bugs Bunny cartoons for examples). Thus it can be hard for North Americans to pick it up -- a number of websites offer tips on how to learn to roll your Rs.
"The English R is really different, so it gives you away immediately when you use it," says language teacher Benny Lewis.
As you say, though, it's close to our 'T.' A few sites (including Lewis's) recommend "faking" a rolled R by replacing it with a soft 'T' sound (as in 'water').
Fieri is a popular, and busy, Food Network personality whose other shows include "Guy's Big Bite," "Guy's Grocery Games" and "Rachael vs. Guy: Celebrity Cook-Off." However, it all started with his first restaurant, the Italian-themed Johnny Garlic's.
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