Strategically: I've only ever see it work by making it somebody's job to save money on Amazon, ahead of feature velocity. Find a tool that analyses where you're spending , pick the biggest bucket of spend you don't really understand, and drill down until you're sick of looking at it. Make some optimizations around how you're spending that money, rinse, repeat.
Every org I know has done a first pass that's pretty effective, where you buy a bunch of reserved instances for the compute power you need. A couple of the companies I've worked with stalled out at that point. The others figured out real cost / benefit models for real projects and did things like move data from S3 to Glacier, or from hosted MySQL to RDS, or from Cassandra to DynamoDB.
There are only so many free-lunch cuts you can make, so it's worth your while start down a path that allows you to consider trade-offs, which includes empowering somebody to lobby for them.
 Seems nice, haven't used it: https://github.com/Teevity/ice. I know the folks at http://cloudhealthtech.com, they're building a pretty solid business if you're willing to spend money to save money.