I am fairly bored myself, and thus, here is a short-ish list of anime I enjoyed, along with an extremely condensed summary or explanation.
EDIT: In my haste to recommend things I like, I forgot to mention something. I find it fairly silly to dismiss an entire form of media based on a few experiences. We've all seen bad movies, read horrible books and played awful games, but I rarely see people claiming all movies/books/games are worthless. Anime is just another form of media. Some is bad, some is mediocre, some is good and some is fantastic.
Princess Tutu: A fairy tale written by an author with magical talent comes to life and begins to affect a small town. The author magics a duck (yes, an animal, think Swan Lake) into his new heroine in order to move the story forward. This anime uses ballet as a central theme and all the music is classical. Note, occasionally very obviously low budget and has a lot of contrived dialogue because it's a fairy tale, but one of the best stories I've ever watched. This show does seem extremely girly- please don't dismiss it based on that alone.
Baccano!: Set in the 1930s, this is the story of a bunch of gangsters, bootleggers and general criminals (some secretly immortal thanks to alchemy, some not). Mostly deals with how each character weaves into the story and how the same event is different depending on how you see it. Fantastic music and animation, amazingly likeable characters. Note, the story is told in an extremely disjointed way, so first-time watchers will be VERY confused for several episodes. Stick with it, it's worth it. Warning: Do not watch this show if you are sensitive about blood, gore, and violence. Highly recommend the English dub since it's set in the American east coast.
Panty & Stocking with Garterbelt: Absolutely insane show about naughty angels who must fight ghosts to earn their way back into heaven. Completely, unashamedly immature, and extremely enjoyable to watch (for me anyway.) Bright, colorful art, fluid animation, awesome techno soundtrack.
Haibane Renmei: Mostly metaphorical show about a group of girls in purgatory and coming to terms with death. Worth watching even though there's no effective way to promote or explain it without spoilers.
Fullmetal Alchemist: Two young boys attempt to bring their mother back to life with alchemy. This ends horribly with both of them permanently maimed (one requires an artificial arm and leg, the other loses his entire body and ends up as a soul bound to a suit of armor). They go on a quest to find the Philosopher's Stone in order to fix their bodies and get involved with the military in order to gain access to classified information.
Azumanga Daioh: Follows a group of girls through their years at their Japanese high school. My friends' pet name for this show is "Awkward Pauses: The Anime." A slow-paced but quite hilarious comedy show. Based on a series of 4-panel comics, so don't expect any big story arcs or anything. This is just for silly laughs.
Cowboy Bebop: Three bounty hunters (one ex-gangster, one ex-cop, and one con-woman), a dog and a strange, computer-genius girl all live on a spaceship together and take on bounty jobs. Fabulous music. This show is a staple in pretty much all anime fans' collections; it hardly needs explanation.