I usually check email in the evening before going to bed. It's working marvelously since I can take an hour out of my day to handle everybody - really solves the problem of forgetting to respond to people. The only problem are time-sensitive emails. For those I still need to periodically check my inbox to see if something urgent's come up. If there's a way to tell Tray "If time-sensitive email, send me a tweet", that…
There's just so much to it that it seems like something only a human could do. That said, the guy from tray.io who replied with a potential solution seems to be on the right track. Its not a perfect solution but it's good enough and sometimes good enough is all you can do.