I can't speak for the accuracy of the hanzi translations in the first answer here, but as I looked at them, some seemed to be a bit 'off the mark'. So here are the meanings in English ...
1. 'wear your heart on your sleeve' -- to show your emotions and feelings with body language, facial expressions, and spoken words; you don't try to hide what you feel in your heart.
2. 'ever-popular' -- popular in the past and still popular now
3. 'chicken-hearted' -- cowardly; not brave at all. However, in spoken English the word 'chicken' is usually used by itself. You hear it mostly from young children (especially boys) making fun of someone, or trying to goad someone into doing something they are afraid to do; for example: "Come on! Jump! Are you chicken?"
4. 'heart is in your mouth' -- Here in the US, we say 'heart is in your (my) throat', but apparently 'mouth' is the form used in the UK. It means frightened, or extremely nervous about something you must do (e.g. make a speech), or something that is about to happen to you.
5. 'put your heart at rest' -- a more emotional way of saying "don't worry anymore". It wouldn't be used for small, day-to-day worries, but rather for more important, personal concerns (e.g. when you tell your mom you passed the 高考). I always thought 放心 was a good Chinese equivalent.