After posting the the excerpt from Arkansas' on Brian's comments section, I decided to go one step further- since I'm just sitting here watching television, anyway.

Arkansas:
Constitution Of The State Of Arkansas, Article 19. Miscellaneous Provisions: 1. Atheists disqualified from holding office or testifying as witness. No person who denies the being of a God shall hold any office in the civil departments of this State, nor be competent to testify as a witness in any Court.

Maryland:
Maryland's Declaration of Rights, Article 36
"That as it is the duty of every man to worship God in such manner as he thinks most acceptable to Him, all persons are equally entitled to protection in their religious liberty; ... provided, he believes in the existence of God, and that under His dispensation such person will be held morally accountable for his acts, and be rewarded or punished therefore either in this world or in the world to come."

Mississippi State Constitution. Article 14 ("General Provisions"), Section 265
No person who denies the existence of a Supreme Being shall hold any office in this state.

North Carolina's State Constitution, Article 6 Section 8
"Disqualifications of office. The following persons shall be disqualified for office: First, any person who shall deny the being of Almighty God."

South Carolina's State Constitution, Article 4 Section 2
"No person shall be eligible to the office of Governor who denies the existence of the Supreme Being; ..."

Tennessee's State Constitution, Article 9 Section 2
"No person who denies the being of God, or a future state of rewards and punishments, shall hold any office in the civil department of this state."

Texas' State Constitution, Article 1 Section 4
"No religious test shall ever be required as a qualification to any office, or public trust, in this State; nor shall any one be excluded from holding office on account of his religious sentiments, provided he acknowledge the existence of a Supreme Being."