Although I only give this 6.5/10, I really did enjoy watching Kamisama Hajimemashita and would love to see a sequel that goes into more detail about the previous shrine owner and what his motivations are.
Story: Nanami loses the house she lives in after her father disappears. While contemplating her situation in the park, she saves a treed man from a dog. He gives her his home as a thank-you gift, but that home turns out to be an abandoned shrine. Nanami must work to gain the trust of Tomoe, the shrine's pretty main familiar with a tragic past, and you have the beginnings of a supernatural love comedy.
Love comedies require a tricky balance between comedy and romantic development, and I think this one does a good job. It's not mopey or melodramatic, nor is it sappy. In fact, Tomoe and Nanami spend most of the show comically arguing with each other. The climax of the last 2 episodes is a fairly predictable "love conquers all," though, so don't expect to get anything new or revolutionary out of this show.
Animation: Big shoujo eyes and skinny bishounen everywhere. Typically, the characters get more attention than the backgrounds, but there's really not much to complain about here.
Sound: I liked the voice actors, and the light Jpop theme songs. They fit the mood and setting of the show.
Characters: Did I mention bishounen everywhere? There are two main and a handful of side characters, but the only one who showed any kind of character development was Tomoe, and even then it was really only him begrudgingly accepting that he loves Nanami. The main character herself was similarly disappointing: too boring, too weakly characterized, with not enough of her own personality, hopes and dreams.
I think anyone with a taste for shoujo will eat this show up along with its bishounen. Other than the fairly weak characters, it doesn't contain any major flaws, but I would not recommend it to anyone who dislikes shoujo.