Unfortunately, Echo only implements on-device keyword spotting to detect the "wake word". When the device detects the "wake word", it streams audio to the cloud.
Thus, the "wake words" such as Alexa, Amazon are inherent to the Echo device. The Amazon Voice Service (AVS) has no "wake word" implementation. Same thing for the Amazon Skills Kit (ASK). So, the only options are to implement a physical push button as indicated in the RPI project or integrate a third party speech recognition package such as Sphinx
http://cmusphinx.sourceforge.net/ with the RPI project. The 3rd party package would "listen" for the "wake word" which would in turn activate the AVS "listening service" upon "hearing" the "wake word".
Using the latter option, gives you more flexibility in implementing a customized "wake word". For example, you can say "PI, turn on kitchen lights".