Per my annual tradition in the days (or in this case the hours) before Yom Kippur (literally "The Day of Atonement"), I would like to ask your forgiveness for any offense or transgression I might have committed against you in the past year (background for my non-Jewish friends: http://www.jewfaq.org/holiday3.htm). Last year my former boss, who is Indian and not Jewish, called this "emotional defragmentation." I thought this was a great way to express the idea behind Yom Kippur in non-denominational terms.
If telling me what you are forgiving me for - or what you would like me to apologize for - aids the catharsis, all the better. Just please send me a personal note rather than posting on my Facebook wall!