If you are looking for a clean, modern LMS that is already integrated with Google Apps, I would highly recommend Haiku. Haiku has a beautiful interface with simple drag and drop features. It feels great, is web-based, and is just what a LMS should be. Haiku is a high end LMS at a very small price.
If free and simple is what you are looking for then check out Edmodo or Schoology. Both have a Facebook-like feel and are perfect for teachers looking for an online space to post projects, have student collaborate, and develop a foundation of digital literacy in the classroom. Lastly, both platforms have developed apps for Android and Apple mobile devices.
Still looking? More LMS resources can be found here to help you learn more.
Bonus: The next Power Searching MOOC with Google is starting on September 24th - Check it out!