Show the user a dialog after 3 days of use (I additionally make sure that the app has been opened at least 5 times) asking him or her to review your app on google play if he enjoyed using it and you will upgrade his app to the pro version for free.
You can't check if the user actually reviewed your app, or if the user gave it a good or a bad review, but I had good experiences with that. That's a simple mechanic that will get you some nice reviews, because a user that regularly uses your app after 3 days probably wont hate your app. :-)
You can easily compare the result of this by checking this link to my app:
https://play.google.com/store/apps/details?id=com.flipster
We are both in the same category of downloads (between 1k and 5k - I currently have 1,7k downloads) and its 47 reviews vs 8.