If you have a look at Cicero's letters, many of them do not have any valediction at all. In a pair of letters exchanged between Q. Metellus and Cicero (Cic. Fam. 5.1-5.2), the two men simply stop and end the letter without any closing.
However, there were common ways of providing a common valediction. One of the most common you can see at the end of Cicero's fifth letter to his friend Atticus: cura ut valeas, "take care that you are well." This was a standard valediction not limited to Cicero. It also can be expanded, as you see he goes on to urge Atticus' continued friendship with Cicero and his cousin.
Other times, a simpler vale (good-bye) is used instead. In a letter to Gaius Memmius (Fam. 13.1), he ends the letter simply with vale, but in a letter to his wife and daughter (Fam. 14.2), Cicero signs off with valete, mea desideria, valete, "good-bye, my loves, good-bye."
It's further interesting to note that Cicero actually ended that letter with the date and place in which it was written. Sometimes you see this at the top, other times at the bottom, though much of the time it's absent.
Another interesting variation on vale is etiam atque etiam vale, where etiam atque etiam "denotes that an action is done uninterruptedly, incessantly; whence it also conveys the idea of intensity." (Lewis and Short)
Among the common people and later than Cicero, "I hope that you are well" became a common valediction, represented by either valere te opto or valere te cupio; sometimes bene was included for strengthening the wish. There is some disagreement, though, about the origin of the phrase, whether its native Latin or borrowed from the Greek, though J. N. Adams in his Bilingualism and the Latin Language (p. 79-80) claims the Vindolanda tablets should indicate Latin priority.