Employers can consider many factors in deciding whom to hire. Restoring your "rights" does not wipe the slate clean on a background check. Restoration of legal rights allows you to do things previously forbidden, but does not guarantee you a job.
Restoration of civil rights does not remove the fact that you were convicted of the felony. Background checks will still show it, restoration of rights is not an expungement. You can certainly explain that the convictions were very long ago and that your rights are now restored, but absent a pardon from the Governor, your record will reflect the convictions.
The criminal cases against you are a public record, and they will be forever. Consult with a local attorney about an expungement or perhaps a pardon. Convictions usually cannot be expunged.