Start service error code 3417 with SQL Server 2012
That error is usually associated with the Master database, so you are on the right track. Does the service account have access to the directory where the Master database resides?

There should be some additional errors in either the SQL Server error log or the Windows event log that may narrow down the issue. Look for access denied messages or cannot find file messages associated with the master database and other system databases. 

One other thing I have found on other sites is folder compression being a problem. Make sure SQL Server does not reside on a compressed folder or drive.

If that doesn't work, then rebuild as suggested.