Devices like the Kindle Fire are Android-based and don't connect to gApps, so Google is not a requirement. As others have said, you can use GetJar, Amazon Appstore, and other sites to get applications. There are a few apps that will give you email without using gmail, and the phone should have a local calendar that can be used.
The only thing will be not being able to use Gmail, Google Calendar, Google Marketplace, and any other Google apps that require logging in.