Fluent.io by syncing it with your Gmail account. The interface looks like a lot like the Twitter app for Mac, where your dashboard navigation is on the left and you can toggle between your inbox, to-do list, emails with attachments, all your mail and your list of labels.
All items in your inbox are displayed in a stream of short messages that you can easily apply actions to with one click. Replying to an email has the same design interface as a Twitter message or Facebook wall. You can quickly reply to all people in an email within the stream, or customize your message by clicking "change" and a separate window will pop up that looks more like an email message layout. Archive emails, turn them into "to-do's" or label them all with simple buttons at the top of each email message.
Our biggest complaint with Gmail would be the confusing layout when a lot of people are responding to one large email chain. We like how Fluent.io solves that problem visually with its message-like design. Fluent.io is built based on the reality that most of us are checking email on a pretty constant basis depending on how our notifications are set up. We respond to emails almost automatically, especially during work hours and sometimes send short messages back and forth. Email has evolved more into short messaging and less like physical letters. However, Fluent.io is currently optimized for desktop use, not mobile. You can give Fluent.io a test drive via their demo site to see if this UI fits your workflow.