Blossom viewing is common to all East Asian cultures, and the cherry blossom is the most prominent spring blossom in Korea. I would not say it is central to the culture the way it is in Japan, however.
As with everything in East Asia, it is not untouched by complicated history. Some of the more famous stands, such as at Changgyeonggung Palace in Seoul, were planted during the era of Korea's occupation by Imperial Japan. Both China and Korea claim to be the origin of Japan's cherry trees. But I don't want to overstate things. No one wants to talk politics when they're out enjoying the blooms. Indeed, sakura, used as a loanword (사쿠라), is the most common way to refer to the flower (the native Korean word is beot-kkot, 벚꽃), and the activity of blossom viewing, similarly, uses the loanword hanami (하나미), natively kkot-gugyeong (꽃구경).
The biggest cherry blossom festival is the Gunhangje Festival, which runs for ten days in the early spring in the Jinhae district of Changwon, a seaport and the capital of South Gyeongsam Province (Gyeongsamnam-do). Jinhae is planted with hundreds of thousands of cherry trees, and during the festival there are parades, performances, and other cultural events, such as commemorations of national hero Admiral Yi Sun-Shin (Jinhae is home of the naval academy and a major naval base; gunhang means naval port). The Korea Tourism Organization has a listing of the most famous cherry blossom sites in Jinhae, notably the "cherry blossom tunnel" along Yeojwacheon Stream.
On the other side of the province is the Simni Cherry Blossom Road, a 6km walk in Hadong leading to Ssanggyesa Temple lined with cherry trees. The path is nicknamed Hollae-gil (wedding path) because of a legend that couples who walk the path together holding hands will be together for one hundred years.
The Jeju Cherry Blossom Festival in Seogwipo also runs in the early spring, with ceremonies, concerts, street festivals, and other cultural events. As this is the southernmost populated part of Korea, it is the first part of the country to see cherry blossoms.
If you cannot make it out to the provinces, there are plenty of cherry blossoms all around Seoul. The most famous viewing spots are Yeouido Park, along the Namsan Circular Road, at the main entrance to Kyunghee University, and at the aforementioned Changgyeonggung Palace.
But there are other blossom festivals in various parts of Korea, notably the plum blossoms (Gwangyang International Maehwa Festival), Japanese dogwood/cornus (Gurye Sansuyu Festival and Icheon Baeksa Sansuyu Festival), canola (Jeju Canola Flower Festival), and azaleas (Yeongchwisan Azalea Festival) among others.