Terry was born and raised in Central Louisiana. His parents, the late M.L. Luneau, Jr. and Myrna Burnaman Luneau were also life-long residents of Central Louisiana. Mr. Luneau was associated with the Timber Industry and Mrs. Luneau has worked in the accounting department at Central State Hospital for over twenty years. Terry's brother W. Jay Luneau, Sr. is an attorney in Alexandria.
Terry is married to the former Terri Voorhies of Alexandria; She is part owner of A.P.A.S.C.O., an automotive paint store. They have two sons, Nicolas and Zachary, who are both students at Grace Christian School. He is a graduate of Pineville High School and The Dallas Institute of Funeral Service. The Luneau's lived in Shreveport for several years where he worked as Director of Anatomical Services at L.S.V. School of Medicine.
He is an active member of Calvary Baptist Church where he is the Royal Ambassadors Director and Sunday School Teacher. Terry is also a member of The MacArthur Dr. Lions Club and currently serves as Club President and on its board of directors.