Most people who I know who are studying Japanese or want to learn Japanese are mostly into anime. I try to encourage people to watch dramas but most just don't seem interested. :/ It's a shame, because there are some great series out there. And you can finish them in a fairly short amount of time, since dramas tend to usually be twelve episodes at most, thought this varies. I think most people are not drawn to dramas because they are too reminiscent of soap operas, and tend to have romance heavy plots, lots of tears, and cheesy acting. But hey, so does anime. *AHEM*
While dramas do have their fair share of cheesiness and overly dramatic plot lines (they are called dramas for a reason), there are plenty that don't involve the typical romance, love triangles, terminal illness and cheesy plots. Of course, even the least cheesy dramas have that typical dramatic element, but overall, I think there are lots of dramas that even the most reluctant person will enjoy. I'm not going to base this on ones that I liked, but rather dramas that I both liked and disliked that don't contain the typical cheesy factor. There are many popular dramas that I hated so consider this a completely unbiased list.
Kisarazu Cat's Eye
Nobuta wo Produce
Yankee-kun to Megane Chan
My Boss My Hero
The Quiz Show
Click the titles to be taken to the drama-wiki page. Generally, detective dramas, mystery dramas, cop/crime dramas, dramas involving gangsters and delinquents, and sometimes sports dramas tend to have the lowest amount of cheesiness. Suspense dramas are good to watch also. There really are very few dramas that are free of sappy moments and cheesiness. However, there are many that don't focus on romance and love triangles and overly dramatic plots. If you are really interested in watching them, then you will just have to do your research. Keep in mind that most of the American audience that views Japanese dramas is female, so dramas that feature good looking actors and romance are going to be more popular. Instead of checking whats popular, find dramas based on the genre or actors that are in them.
Dramas are like anime in that there is a drama about practically anything since there are so many made each year. There are so many different types, and many are actually based of manga and anime (though liberties are often taken with the plots). I think that it's worth it to at least give watching dramas a try. I'd say that one that is good to start off with is Gokusen. The author and creator of AJATT himself has praised the series as having greatly contributed to his Japanese. It can be a little cheesy at times, but I think it balances the cheesy scenes with good humor. Plus, it is set in an all-boys school so no love triangles and jealous scheming ex-girlfriends. Two others I would recommend are Bloody Monday and Liar Game. These dramas deal more with suspense and very little with romance. They have logic based plots and leave you hanging on with each episode. My Boss My Hero is great fun as well, and Nagase Tomoya is hilarious. Basically, these are dramas that guys who don't care much for the soap opera-like series.
I definitely think it's worth a try and you will probably find at least one series that you enjoy.