With Father's Day taking place on June 21st, 2009, here is a video from the Mormon Messages YouTube channel that well defines the LDS attitude towards Father's Day. It's entitled "A Father Indeed", and is the story of a committed husband and father who consistently and passionately demonstrates his love and devotion to his wife and kids.
From the LDS Gospel Library, we find out how the General Authorities view the role of the father within the greater context of the family unit:
"The family is ordained of God. Marriage between man and woman is essential to His eternal plan. Children are entitled to birth within the bonds of matrimony, and to be reared by a father and a mother who honor marital vows with complete fidelity. Happiness in family life is most likely to be achieved when founded upon the teachings of the Lord Jesus Christ. Successful marriages and families are established and maintained on principles of faith, prayer, repentance, forgiveness, respect, love, compassion, work, and wholesome recreational activities. By divine design, fathers are to preside over their families in love and righteousness and are responsible to provide the necessities of life and protection for their families. Mothers are primarily responsible for the nurture of their children. In these sacred responsibilities, fathers and mothers are obligated to help one another as equal partners. Disability, death, or other circumstances may necessitate individual adaptation. Extended families should lend support when needed".
The father is called to preside over the family, but is also called to help his wife as an equal partner. This is patriarchy. Done right, there is no system more godly. Done wrong, there is no system more devilish.