Beautiful Life (2000) - one of the very first KimuTaku's dramas I watched, and it was a wonderful romance drama. Japan excels when it comes to depicting daily life and struggles of the disables and minority.
Summer Snow (2000) - another classic romance, imho.
Kimi wa Petto (2003) - imo, this version is better than the newer version. The actings were more natural, Koyuki and Jun are simply better actors. Even when this is an older woman-younger man romance (and and odd relationship to begin with), this drama still grabbed my heart.
Crying out Love, in the Center of the World (2004) - This drama was very popular the year it came out. Be forewarned, this is yet another romance/melodrama like the first two dramas listed above. But it's also very good.
Orange Days (2004) - one of my all time favorites. Besides romance, this drama also focuses on friendship and college life.
Last Christmas (2004) - This is quite an underrated romance drama. It's sweet and heartwarming.
H2 (2005) - an adaptation from Adachi's manga. If you're familiar with this manga author, you'd know that he does subtle romance AND love triangle really well.
Love Shuffle (2009) - another cute one with multiple couples and strong friendship theme.
Nagareboshi (2010) - a very beautiful and touching drama. Now, this one has a strong undercurrent of romance, but it is very subtle. Still, it's one of the best romance dramas out there if you ask me, even when the romance is often time not the main focus.
Hotaru no Hikari is another great one, but this is a popular drama and seems like someone had already recommended you. Enjoy watching!