I have to add another thumbs up to City Hall. Get past the first episode (which so many say is slow, though I didn't think it was), and it'll take you here, there, and everywhere rooting for the everyday heroine. It also has one of the best endings ever.
If you don't angsty shows with tears, but with good endings, try Spring Waltz (which is visually stunning), Boys Over Flowers (it's such a schoolgirl squeal fest, but a good guilty pleasure with a good ending), My Girl, Baker Kim Tak Goo, Kimichi Family (one of the best I've seen in a while), I Really, Really Like You, and Drunken to Love You.
If you're looking for more bittersweet (some crying in the middle, but not a river of tears), but with good endings, try My Girlfriend is a Gumiho (funny with bittersweet), Operation Proposal, Stars Falling From the Sky, or Sweet Relationship.
If you want straight out sweet, without using so many Kleenexes people think you own stock in the company, try God of Study, Skip Beat, Flower Boy Ramen Shop, You're Beautiful, Tamra the Island (historical), Pasta (more mature romantic comedy). Just remember, all dramas are going to have some tissue worthy moments; these just have very few.
Avoid at all costs (I learned these the hard way): 49 Days, What Happened in Bali, Assorted Gems, and Hong Gil Dong (great writing and story right up to the end).
