I tend to read mangas/manhwas, without much discrimination (exceptions primarily being the harem and ecchi genres.), but I'm also working my way towards a sizable collection of anime, the newest hard-copy addition, which is Ouran Host Club (I've had the anime for ages, but I didn't OWN it.)
If by "girly", you mean romance animes as opposed to shoujo, which is the genre supposed to be targeted at girls, not that I have much of a problem reading/watching content from that either.
I've got:
Air
Byosoku 5 centimeter - 5 Centimeters per Second -
Chobits
Clannad
D.N.Angel (sooooo much worse than the manga, to my great sorrow)
Eureka 7
FLCL
Koi Kaze
Kumo no Muko, Yakusoku no Bashou - The Place Promised in our Early Days -
Nanatsuiro Drop (just to give you an idea of how much cute has been packed into this one, here's the opening, and that's merely a fraction.) Onegai Teacher
Shakugan no Shanna (both Seasons)
Everything listed is merely what I've currently got in my possession.
It's primarily a western thing that romance = girly, the Japanese have plenty of romance mangas/animes marketed primarily towards guys, in the vein of Onegai Teacher, Onegai Twins and the like.
TLDR: I have no compunction whatsoever about watching girly animes.