"Milady, mine eyes hath not laid upon any visage so beautiful as yours. 'Tis fairer than the sweetest nosegay."
I get why people in fantasy novels don't usually talk the way we do in modern society. Not only do they lack much of our technology (and a lot of our abbreviations and slang come from the ways our language has changed thanks to technology) but it also would just sound weird to have a couple of noble ladies walk up to each other and say "Hey biatch! Wassup?"
But that doesn't mean your characters have to talk with Shakespearean words and weirdly inverted syntax. It doesn't even mean they have to talk particularly formally, either.
I get the temptation. Everyone in LORD OF THE RINGS talks pretty, even the people who really shouldn't talk pretty. In Victorian novels and 18th century diaries, everyone's diction is high and their words are long. Making characters speak intelligently, with long sentences, big words, and without contractions (and I'm guilty of this one myself!) is a good way to make them feel like they're from another time or another world. It takes them out of that modern context, and yes, that is appealing.
But it is SO easy to go overboard with this. If you'd believe it, there's actually a freakishly-high-diction groove you get into once you start writing fantasy people like that. Trust me, when I was 14 ALL my fantasy characters spoke like they'd swallowed the OED and THE COMPLETE WORKS OF SHAKESPEARE. And a modern reader isn't going to be able to connect with someone who talks like Hamlet, anyway--unless they're actually reading HAMLET. (Even then, some people can't stand it.)
So, how do you keep obvious trademarks of modern diction out of fantasy dialogue without making your characters sound like they're being held hostage by Petrarch? Well, you can keep your diction pretty natural, actually. Eliminating things like "like" and "you know" and "seriously" from speech evens it out really well. Calling parents Mother and Father instead of Mom and Dad is pretty much an instantaneous royalty-ification. Don't use words that only came into popular use in the late 19th century or later--and if you're not sure, google it! (So if you're doing a medieval-esque setting, they wouldn't have had words like analgesic or addicted.)
Keep the slang to a minimum (unless you're not going with the familiar royalty route (more on this anon!) and are writing dockside workers--in which case you need low diction and a LOTTA slang), and if you must use a slang term, make one up! Or, slang is a place where it's okay to steal from Shakespeare or Dickens or whoever (though I do not recommend using 'alack!' as your slang word unless you're writing some kind of comedy.) Words for distance and time passage (like fortnight) are good--they sound old-fashioned, but they can still be used in a totally understandable and not at all awkward exchange of dialogue. "When will you come home?" "In a fortnight." And, hate to bring this up, a lot of curse words have been around for a long time, too.
And yeah, trying to come up with a pithy synonym for some of our slang is HARD. Believe me, I know. I spent forever trying to think of a synonym for "shut up!" and I still haven't come up with anything great. ("Be quiet" just doesn't have the same oomph, and something like "still your tongue" is rather awkward to say.) But my advice is, rather than saying something like "Still thy tongue or I shall cut it out with my sword!" just write around it. Eliminate the need for the use of "shut up" and put in something else--clapping a hand over someone's mouth, socking them in the jaw, snapping at them to make them stop talking, screaming over their words, whatever. (I do not recommend the use of "whatever" to end a sentence in fantasy dialogue.)
The biggest piece of advice I can give you is this: use your ears. You know when something sounds lofty and awkward, just as you know when something sounds modern. If you have to, read it aloud to yourself, or read it aloud to someone else. Your ears will tell you when the dialogue is right.