I highly recommend "Nobuta wo Produce", a friendship drama, and "Buzzer Beat", a sports and romance drama. (I'm not into sports, but it was more of a setting than the focus). "Kurosagi" is a popular one in the crime/suspense genre, and pretty interesting.
"Code Blue", a medical drama, and "Proposal Daisakusen", a romance drama (I think comedy), are very popular but I haven't seen them.
"Saikou no Jinsei no Owarikata" is a serious drama from last spring; it got mixed reviews but I thought it was well done. "Sore wa, Totsuzen, Arashi no Youni" is a wistful romance drama from a few years ago. It's not very well known but I loved it.
You can also check out this article from the Stalker's Guide series:
http://mydramalist.com/article/2012/10/12/a-stalkers-guide-to-yamashita-tomohisa#.UPxbzydEGZg