Thanks @luisfsr! That made me discover Geary, which is really a sweet app. Can’t wait for it to support cryptographic signing and encryption!
If this wasn’t clear from my “nicely integrated” statement: I’d really love to see a GNU/Linux desktop that behaves somewhat like a mobile device, i.e. no need to start up a Mail app (Thunderbird that takes quit some time and resources) and a Calendar app; instead you get notifications just like on your phone about incoming messages and due appointments, and when you click on a notification the related app opens.
The desktop should feel just as lightweight as a mobile device! 