Yes, you can definitely read non-Amazon books on a Kindle. You'll need to obtain the eBooks in a format compatible with your Kindle device or app. This could mean downloading from other eBook retailers, or converting files from different formats.
One way to quickly and easily find titles that are compatible with Kindle is to use the Kindle MatchBook program. This program offers eligible print book customers the option to purchase Kindle editions of those titles for a discounted price.
Another way is to use third-party apps for your device or app to download eBooks in the right format. For example, apps like Overdrive, Calibre and Azardi offer this option.
Whichever method you choose, it's possible to read non-Amazon books on mobiles, tablets and e-readers!