Spring and fall are two big festival seasons in Jeollanam-do, with a good dozen festivals worth visiting. Pickings are slim in March, and may be even slimmer if festivals are cancelled this month due to foot-and-mouth disease (like they were in December and January), but there are a few big ones coming up.
The annual Gwangyang Apricot Blossom Festival (광양매화축제) will be called the Gwangyang International Apricot Blossom Festival (광양국제매화축제) for the first time in 2011, apparently for no real reason ("광양 매화축제, 이름만 '국제'" ), though domestic festivals are known to do this to sound more, well, international. It will run from March 12th through the 20th.
The Gwangyang Apricot Blossom Festival is usually the first spring festival, and marks the beginning of the season for southerners.
Both of my attempts to visit it were unsuccessful: once because I couldn't find the bus and another time because it was too damn cold. Your best bet is to take a bus to Hadong county in Gyeongsangnam-do and walk through town into Gwangyang. You'll find apricot blossoms along the river and will pass through a small market and festival area before finding the shuttle buses that will take you to the Apricot Blossom Village (매화마을). The official website hasn't been updated since last year. See this post of my half-visit in 2009 for more information.
In Gurye county the annual Sansuyu Festival (산수유꽃축제) runs from March 17th through the 20th. The official site hasn't been updated since last year, though it tells you buses run regularly from the terminal in Gurye-eup.
And, the regionally-famous Jindo Sea Road Festival (진도 신비의 바닷길 축제), known also as the Sea-Parting Festival and the Jindo Moses Miracle, will take place from March 19th through the 21st. A few times a year the tide goes out and allows people to walk the 2.8 kilometers between two small islands in Jindo county. The program on the homepage is still under construction, though you'll want to take care to visit on the day when you can actually walk between the two islands. In addition to wading into the sea there is the sort of food and performances you'll find at every Jeollanam-do festival, as well as Jindo dogs and other entertainment.
Here's a profile on the Sea-Parting Festival from the Korea Tourism Organization, and pictures from Jindo's biggest blogger, Living Life Frame by Frame, here and here. Accommodation will be scarce, so if you're from out of town I recommend you find a motel in one of the nearest transportation hubs: love motels are plentiful outside of the bus terminals in Gwangju and Suncheon, for example.