To me, this is a pretty key feature that has been missing from the Amazon Echo so far. We have been using the Echo as an input mechanism for home automation commands, but the realm of event notifications has always been reserved for smart speakers like Sonos. With this method from the SmartThings forum, the Amazon Echo takes over that notification role, and in my opinion, this really drops the value proposition of expensive smart speakers like Sonos. This also reinforces why I believe it is so important that the home automation devices I buy must have an open API, so that its basic functionality can be extended easily by the community.
What I did was to follow the instructions on this thread: Android as a Speech/Alarm Device released!. The basic idea is to push events out from SmartThings via a "Big Talker" SmartApp, then have the notification played on any spare Android device on the same network as the SmartThings hub, via a "LANnouncer" device type and Android app. The Android device is connected via Bluetooth to the Amazon Echo, and hence the notification gets played out on the Echo. The Echo is intelligent enough to be able to switch inputs between its own music stations, and incoming notifications from the Bluetooth device.
Instructions on how to install LANnouncer are best described in its own install page. While the custom SmartApp code for Big Talker can be found on its own GitHub page. The following screenshot from the SmartThings UI shows how I configured 2 switches to be announced by the Big Talker SmartApp.