I like first-person fantasy, most of the time. I enjoy seeing how authors can use the first-person voice, what kind of explanation they'll craft for how it tells the story, and the fact that first-person mostly locks the author into one viewpoint instead of bouncing like a mad thing all across the countryside (Robert Jordan comes in here for his share of kicking as usual). But there are definitely ways to misuse it, especially when someone who's never told a story before goes for first-person because she thinks there's no other way to do it.
So, some ways I think are good for using the power of the first-person voice while avoiding the common pitfalls.
1) Decide first what mechanism you're using to tell the story. There's a convention that will let you get away with pretending that the narrator of the story is telling things as they occur, just like a third-person narrator, and he or she just happens to have a perfect memory for conversations, or else likes to phrase the common events of his or her life in past tense. Most people won't get on your case about this, though I have heard it cited as one of the reasons some readers don't like reading stories in the first-person.
However, if you pick this convention, stick with it. No flash-forwards or hints of something the narrator couldn't have known at the time. "As it turned out, I shouldn't have trusted him, but I couldn't know that until later" is a no-no with this convention.
It's easier when you're presenting the first-person story in a diary format, or (as Steven Brust does) the tellings of the central character to an unknown listener. Then you can say things like this quote from Brust's Yendi, "I turned away, though, because, odd as it may seem to you who have listened to me so patiently and so well, I really don't like pain." The narrator can more easily jump outside the story and summarize events, give descriptions, or point out the difference between what he knows "now" and what he knows "then" (story-time). The usual marker of this is the present tense. "At the time I lived in Brookstream, it was a small town, but now I suppose it is a much larger one, with bustling havens and shops radiating the faint smell of cinnamon canassi." And so on.
I've read readable, workable stories written in all these formats, some of them among the best fantasies I've read. (Same-time narrator: Carol Berg's Rai-kirah Saga. Diary format: Robin Hobb's trilogies about the Farseers. Telling to an unknown listener: Most of Brust's Vlad Taltos series, Roger Zelazny's first five Amber books). The important thing is to pick a format and stick with it, instead of bouncing in and out all over the place.
2) If you alternate first-person points of view, make them a) clearly separated and b) far enough apart in time to avoid annoying the reader. It is a pain in the ass to read a story from two different first-person points of view and never know, when a chapter switches, if the author has continued with the first character or gone on to the second. (Third-person narratives, of course, usually don't have this problem unless there are several nameless narrators). So make sure some separation is clear, whether it's chapter breaks, putting the character's name at the beginning of a POV section, using flashback italics, or some other method that tells your audience, "We're moving now." Brust's Orca is a joy to read primarily because of the dizzying effects of such a structure, with nested first-person POVS interrupted by interludes in which one of the first-person characters tells parts of the story to the second character's wife. She doesn't reveal everything, though, and so the reader is left to adopt mostly her perspective, though it isn't the only one or even the only first-person one in the book. Orca makes you work to read it, and I find that wonderful.
However, I would have become increasingly annoyed with Brust if he had jumped every few pages or paragraphs. This was the structure of Elizabeth Kerner's Song in the Silence, and it continually chopped apart the action to tell the scene from another person's POV. I put down the book seasick, and have vowed never to read anything she writes ever again because of it (well, that and the sappy romantic language and the flaming Mary Sue heroine). I couldn't take it.
3) Don't make your first-person narrator present at too many secret conversations, meetings, or murders. I can always tell when an author doesn't feel truly comfortable with first-person and is probably more used to infodumping through the omniscient voice, because her character just "happens" to overhear more plot points than you can chase down with angry sheep. Not the way to go. Part of the fun of having a first-person narrator is keeping him or her in the dark, and letting him or her go after the plot instead of forcing it all to come along. We can often follow a first-person narrator's thoughts more easily than a third-person narrator's, and so readers can have the triumph of coming to a correct conclusion along with the narrator. It might not work that well with mystery plots (Brust makes it work, but few other authors do), but a typical high fantasy/court intrigue plot could work wonders with it.
Of course, the first-person narrator shouldn't be at the edge of all the action and hear it reported too many times afterward, either. In that case, what you probably want is to make the person at the center of the action your narrator, unless you plan to give your minor character a truly interesting perspective and make the story as much about his observations as about the events of the plot.
4) Don't give your narrator all the best lines. It's easy to wind up wanting to strangle a first-person narrator who's too smart-aleck, especially when she's quite obviously the author's darling and stuns people into silence with comebacks that aren't all that witty. The first-person narrator shouldn't float along in a little protective bubble just because she's less likely to die. In fact, it's very effective to start writing the story in first person, make your audience relax with thoughts of, "Oh, he's telling the story, he can't die," and then dump a whole bunch of shit on him to make your readers jump in shock. Carol Berg does this very well in the Rai-kirah Saga, where Seyonne suffers and suffers, but never turns self-pitying.
The best technique for avoiding both self-pity and smart-aleckry, I think, is to surround your narrator with fully fleshed-out characters who can smack the character out of his whining or have wit for themselves. Just about every first-person writer in fantasy I like does this, and their deaths or suffering can be just as affecting as the death of the narrator in another book. It also helps combat the "Author's Darling" Disease, where the narrator can get away with anything because the author is drooling at her feet.
5) Long descriptions of beauty sound even worse in a first-person POV than a third-person POV. Number one lesson in character description for any way of writing should be, "Do not park your character in front of a mirror and let him look at his own reflection," but it goes double for first-person narrators. The author might- might- get away with saying, "She had eyes like stormy oceans, and waves of deep golden hair flowing down her back like cornfields, and her cheekbones were high enough to enchant the most wandering eye," though it would make me break out in hives, but make that character describe herself and see what happens. "I had eyes like stormy oceans, and waves of deep golden hair flowing down my back like cornfields, and my cheekbones were high enough to enchant the most wandering eye." Eergh. Eeergh. Ick. No.
If you don't spend part of the book in the viewpoint of another character who can describe the narrator from the outside, then the best way to do it, as far as I'm concerned, is surround the narrator with people who have some of the same characteristics and slip in bits of information here and there. "She had blue eyes like mine" would be an example, or "Like all our people, he had wings as wide as the span of his arms." Again, don't overdo it, or the reader will be led to the (rightful) suspicion that you've only created the other character as a more sophisticated mirror tactic. Split it apart and let what's necessary pass when it is, and readers can build up a complete picture of your narrator without having to listen to her rhapsodize on about her own beauty.
6) The narrator has to be interesting enough to make us like her. It would seem that a first-person narrator would have the least trouble with this of any character in fantasy, but authors are prone to let clichés they would attack elsewhere slip in when they're part of a first-person narrator's thoughts. They might scorn writing an angsty monologue that goes on for ten pages about how bad their third-person narrator's life is, but a first-person narrator will mentally moan on and on and on if you let her, especially in those scenes with no one else there. Or she might snap a bunch of clichéd "feminist" dialogue at her parents who want her to be a lady, and because the author is counting on reader identification with the character, the writer lets it skate right past.
Cut down on this. Your first-person narrators don't have to be intensely likable, but they have to be interesting; out of the good ones I've already mentioned, I would say that Seyonne is the most likable, and even he makes mistakes. Fitz acts rather weak at times, is stubborn and sullen, and doesn't like apologizing. Vlad is an assassin, mob boss, and pimp who at first sees nothing wrong with killing other people so that he can live comfortably. Corwin has a temper and sacrifices other people's lives to get close to his goal, the throne of Amber. On the other hand, Elizabeth Kerner's main heroine, Lanen, is psuedofeminist, wants Adventure, has Special Mental Powers, describes her own beauty, falls in love in a few days, almost gets sacrificed to demons, and is angsty and woe-sy, and despite sharing her viewpoint for a good portion of the book I was not amused.
I think more people should try first-person, or at least give those books written in it a chance. They can be bloody awful, yes, but when done right, they're often more fun than "epics" that hop from pointless character to pointless character just to show you how big the cast is.