Mecha Anime For People Who Like Characters That Aren't Whiny
Patlabor - Slightly unhinged, perhaps, but mostly just ordinary people doing an honest day's work as public servants [even if they are not well thought of by the public and generally cause accidental collateral damage]. Mostly a light-hearted franchise, especially in the second OVA series. The 3 movies are generally much darker and more serious minded, basically Mamoru Oshii being Mamoru Oshii. Good for cleaning a palette dirtied by emo pretty boys and half-naked lolis.
The Big O - Without getting into the minute details, this is basically Japanese Batman [who happens to pilot a giant robot]. The story centers around Roget Smith, a suave playboy who helps people with problems and usually resorts to using his giant robot when things inevitably get out of hand. Highlights include lots of standalone scifi stories, a retro artstyle, and a soundtrack that's just spectacular.
Gurren Lagann - A series that perfectly encapsulates what first drew people to Gainax all those year ago, but without a whiny emo lead. It's loud, it's vibrant, it's totally over-the-top. Aside from the 27 episode series, there's also two recap movies.
Dai-Guard - The English tagline was "Officer workers saving the world," which is a pretty good summary. The three pilots of Dai-Guard aren't anyone special, they're not even military, they were three people who were picked by a corporation to watch over a giant robot that was built as a marketing gimmick years ago. Unfortunately, giant monster appear and they're the only people that know anything about piloting the robot. The leads are likable, ordinary people, with the main pilot being an excitable guy who just really loves giant robots.
Escaflowne - Someone got their shoujo in my mecha! Yeah, that's what this one is, it's a giant robot series set in a weird fantasy world and this girl gets pulled from the real world into the fantasy world and falls in love with this wild guy who pilots a giant robot [and this other really handsome guy that maybe also piloted a robot, I can't remember, anyone he was SO HOT]. Anyway, it's a Shoji Kawamori x Yoko Kanno joint [with Shinichiro Watanabe doing storyboards and Kazuki Akane handling direction], so you know it's gotta be good. And it is really good. Stay away from the movie, though, since the whole reason you're interested in this list is to get away from the emo.
Martian Successor Nadesico - The main character is reluctant, but I wouldn't call him emo. He just wants to be a cook, not a giant robot pilot. He's also an anime love, who regularly watches show-within-the-show Gekiganger 3, which is a clear homage to the giant robot series of the 70's. It does have its more series moments [especially in the sequel movie], but it's generally a much more light-hearted affair than certain other giant robot shows, and the oddball crew is perfect for rooting for.
Mobile Fighter G Gundam - This is not the Gundam you're familiar with. This is over-the-top, ridiculous Gundam, made abundantly clear by the giant robot that rides a giant robot horse that is actually piloted by a real horse. Not to be taken seriously.
Star Driver - The most fabulous mecha anime ever made.