From R' Gil Student at Hirhurim -
http://hirhurim.blogspot.com/2005/07/shaking-hands-with-women.html:
II. Shaking and Enjoying
Is there sufficient pleasure in shaking a woman's hand to render it prohibited, even according to the Shakh? R. Moshe Feinstein repeatedly (Iggeros Moshe, Orah Hayim vol. 1 no. 113; Even Ha-Ezer vol. 1 no. 56) ruled that a man enjoys shaking a woman's hand and it is therefore prohibited.
As someone who has been shaking women's hands for years, I simply cannot understand his reasoning. A quick handshake yields no pleasure, and is so quick that it cannot even warm a cold hand. It also indicates no closeness between the two parties and is the equivalent of saying "Hello." R. Joseph B. Soloveitchik is quoted as having permitted shaking a woman's hand when necessary, and I once went to the trouble of confirming this from people who asked him directly. R. Mordechai Willig recommends what he calls a "dead fish" handshake -- shaking with a limp hand.
Granted, all agree that someone who does feel pleasure from a handshake should refrain from doing so. But a simple handshake does not, normally, give any pleasure.
I would add that this certainly varies based on culture and in some places a handshake might, indeed, be intended to demonstrate affection or closeness. In that case, it would also be forbidden. In this vein, R. Yosef Hayim of Baghdad (Od Yosef Hai 1, Shofetim) permits a woman kissing a man's hand but not shaking it, because in his culture the latter was considered an affectionate embrace.
I once heard a rabbi, who had previously worked for the OU, speak on this issue. He said that he could not understand why many of his former colleagues would shake hands with women while he went to amazing lengths to avoid shaking hands with women on the few occasions in which it arose. Had he only bothered to ask his colleagues, some of whom can learn circles around him, he might have recognized that (they and) their rabbinic mentors permit this when necessary. (NOTE that this is not intended to imply an OU policy on the subject or that everyone at the OU shakes hands with women.)