One of the two sons of the late Henry and Alice (Buckley) Cunney, Henry Cunney was born in Stoneham on June 4, 1934, but raised in Salem a scion of a well-known Salem family. His father was one of five brothers -- Msgr. Louis I. Cunney, late pastor of St. John the Evangelist, Wellesley and Doctor John Cunney a highly respected physician in Salem to name two of Father Cunney's paternal uncles. Father Cunney's brother Vincent predeceased him.

He attended Salem schools before going to high school in neighboring Lynn, a member of the 1951 class of St. Mary High School there. Following high school he entered the archdiocesan seminaries and on Feb. 2, 1959. Richard Cardinal Cushing ordained him and half his classmates at Holy Cross Cathedral; the other half were ordained at Holy Name, West Roxbury by Auxiliary Bishop Jeremiah Minihan. His was the first class, well half of the class, to be ordained by the recently minted Cardinal Cushing. Boston's archbishop had been named to the College of Cardinals the previous December by Pope St. John XXIII.

During the next 49 years of active priestly ministry, Father Cunney would serve in nine parishes as either an assistant, an associate, or a parochial vicar: St. Peter, Gloucester (1959-1961); St. Andrew the Apostle, Forest Hills (1961-1963); Blessed Sacrament, Walpole (1963-1964); St. Agatha, Milton (1967-1969); St. Joseph, Wakefield, (1969-1970); St. Agnes, Arlington (1973-1975); St. Mary of the Hills, Milton (1976-1986); Sacred Heart, Roslindale (1987-1991); St. Richard of Chichester, Danvers (2001-2008). Sprinkled among those assignments were three assignments as Campus Minister or Chaplain: Our Lady of Nazareth Academy, Wakefield (1964-1967); Bishop Fenwick High School, Peabody (1970-1972) and Regis College, Weston (1986-1987). He was also pastor of two parishes: Nativity, Merrimac (1991-1997) and St. Thomas Aquinas, Nahant (1997-2001). He could rightly brag that he had served in all five pastoral regions of the archdiocese.

Following his retirement in 2008, he lived initially in his own residence and later moved to Regina Cleri Residence. Through most of the dozen years of his retirement, he assisted at various parishes and was a frequent celebrant of the Mass broadcast on Catholic TV.

Eight years into his retirement, Father Cunney penned a reflection of his priestly life and ministry in the July 1, 2016, issue of The Pilot. Looking back then at 57 years of priestly life he still exuded enthusiasm and joy in his vocation. Recalling the various parishes and ministries he had served, he related his admiration or brother priests, especially classmates and for them many friends he had made over those years among priests, religious and laity with whom he had served.

Funeral Services were celebrated for Father Cunney at St. Mary Cemetery, Salem on April 7, 2020, where he was buried in the Cunney family lot.