Well, Chris Tomlin has his own verson of the song Amazing Grace. Its called Amazing Grace, My Chains are Gone! There is a modern remake (not sure by whom) of the song When I Survey The Wondrous Cross! I don't think you are going to have lots of luck finding many church songs set to updated music though, I just don't think its possible. Ceadermount Kids have albums that do songs that are from your listed favorite Christian artist and they do some of the Hymns and other songs you learned growing up like Jesus Loves Me. You may find an album by them that has church songs done up with a nice beat. You might also like Micheal Card, and Ray Boltz they were more popular in the 80's and early 90's though!
I hope this helps!