A handshake is a globally widespread, brief greeting or parting tradition in which two people grasp one of each other's like hands, in most cases accompanied by a brief up-and-down movement of the grasped hands. Using the right hand is generally considered proper etiquette. Customs surrounding handshakes are specific to cultures.
It is indeed a dilemma many of us face. Some Muslims have resorted to showing too much leniency in this area, and saying we need to get with times and shake hands with those of the opposite gender who are non-mahram. One of the misunderstandings is that it is a matter of cleanliness, where the female is seen as "unclean", and hence the male should not shake her hand. Another is that it is a matter of sexualising the shaking of hands.
As Muslims we know our boundaries when it comes to non-mahrams, and it is natural for us as well. We just need to have a sense of pride and respect our Islamic culture and religious laws. We also know that if we are in such a predicament and wont be able to explain or might result in distress, or have time to decline, then it is permissible for us to shake hands.
I have a video tutorial on this issue which might be useful, where I give alternative ideas when dealing with this.
With prayers for your success.
I tend to smile, put my hand to my chest, and say 'Sorry, I don't shake.' By placing your hand on your chest, you are offering an alternative way of greeting, which makes greeting less awkward. Another method is to get a sympathetic colleague - if you have one - to inform people ahead of meeting you that you don't shake hands with the opposite gender.