Not Just Annyeong
In English, there are different ways to greet someone at different times of the day. We say good morning, good afternoon, and good evening. In Korean, we say the same thing to someone no matter what time of the day it is. People generally greet each other 안녕하십니까?, 안녕하세요?, or 안녕? depending the level of speech required.
But that is not all there is. Koreans use other expressions, which may not be similar with that of English, but can be regarded as such. For instance, you would hear a young man greets his grandmother 안녕히 주무셨어요? in the morning. Or a friend greets his friend 잘 잤어? (Did you have a good sleep?) Aside from that, it is also common to hear 밥 먹었어(요)? or its extremely polite form 진지 잡수셨습니까? when someone is more concern whether the person has eaten or not. It is not really odd to ask these things in English, but we can think of them as a more natural way to say good morning, good afternoon, and good evening.
As for good night which is used when people part ways at night or before someone goes to bed, there is a separate expression for that in Korean. You can use any of the following depending on your audience- 잘 자, 잘 자요, 안녕히 주무세요/주무십시요. Nowadays, since more and more people speak English, new expressions are formed such as 좋은 아침입니다 (good morning) and 굿밤 (good night). They are rather literal translation of the English counterpart. So do you greet people in your country the English way or the Korean way?