just off the top of my head:
quakers
menonnites
amish
im sure there are more but i dont feel like looking them up.
heres some examples though:
The 1780 Constitution of the state of Massachusetts, for example, provided that: when any person shall be of the denomination called Quakers, and shall decline taking said oath, he shall make his affirmation in the foregoing form, omitting the words "swear" and inserting, instead thereof, the word "affirm," and omitting the words "So help me God," and subjoining, instead thereof, the words, "This I do under the pains and penalties of perjury."
Conversely, the 1776 Delaware and 1777 Vermont constitutions did not restrict affirmations to Quakers. Like the Federal Constitution, these states allowed any citizen otherwise qualified for public office to affirm loyalty to a state, if conscientiously scrupulous of taking an oath. Still, the federal Constitution went far beyond the practices of even these states by prohibiting religious tests for public office
Joseph Story, an early Justice of the Supreme Court and the author of the first detailed commentary on the United States Constitution, comments on the oath or affirmation clause of the Constitution as follows:
Oaths have a solemn obligation upon the minds of all reflecting men, and especially upon those who feel a deep sense of accountability to a Supreme being. If, in the ordinary administration of justice in cases of private rights, or personal claims, oaths are required of those, who try, as well as of those, who give testimony, to guard against malice, falsehood, and evasion, surely like guards ought to be to be interposed in the administration of high public trusts, and especially in such, as may concern the welfare and safety of the whole community. But there are know denominations of men, who are conscientiously scrupulous of taking oaths (among which is that pure and distinguished sect of Christians, commonly called Friends, or Quakers,) and therefore, to prevent any unjustifiable exclusion from office, the constitution has permitted a solemn affirmation to be made instead of an oath, and as its equivalent (Joseph Story,
Commentaries on the Constitution, 1833, pp. 1838ff.).
http://candst.tripod.com/tnppage/arg11.htm
Many oaths of office contain the text “I, ________ do swear (or affirm)...” When people take such an oath, they have the option of choosing between swearing and affirming, depending on the region where they live. When someone opts to affirm rather than to swear, the oath is more properly known as an affirmation. The difference between the two may seem subtle, but to some people, it is extremely important. It also continues to be an issue in some regions of the world.
Some Christians prefer to say “I affirm” rather than “I swear” because of a section in the Book of Matthew, in which Christ is said to have specifically advised His followers against swearing. Quakers, Mennonites, and members of some other Christian sects choose to not to swear because they believe firmly in telling the truth at all times, and feel that swearing to tell the truth goes against their religious values because it suggests that they might lie at other times.
http://www.wisegeek.org/what-is-the-difference-between-swear-and-affirm-in-an-oath.htm
so, can we lay that tired old dog to rest for once and all?