[Akito Kinomoto]

I think this might be better suited for the Suggestions board since you so ask for anime where the main character's weapon isn't a sword. Anyway at the top of my head:

Magical Girl Lyrical Nanoha (some of them use sword-like weapons though)
Gunslinger Girl (obvious enough)
Tiger and Bunny (my poor recollection tells me some of them might have used an edged weapon but no sword)
s-CRY-ed (Ryuhou uses something like a spear later on)
TTGL (haven't seen but the main weapon here is a drill)

Also because everything is cooler with swords.