Welcome to the Asana Community Forum @Carol_Hackett 
I recommend using the filter. If you want to see unread messages only enable the filter
Also if you ever want to mark a message as unread again you can do so by clicking on the 3 dots and selecting “mark as unread”
You can also add important messages to bookmarks or messages you cannot attend to right now you could create a follow-up task for example to keep the inbox clean.
I hope that helps 