Why hasn't anybody, here, consulted a non-Jewish friend - or a Jewish person with a different belief system, at least - about how he or she would like to have their handshake declined? I would prefer a quick, "I can't shake hands, but nice to meet you" accompanied by a friendly smile.
Don't lie: "I'm sick." (Seriously? Why sin in order to observe a religious mandate?).
And, don't leave my hand hanging in pregnant silence, either. I am not a mind reader, so I had no idea that you don't shake hands when I offered mine for shaking - a very common and generally accepted custom in the US, even among many Jewish people. Be a mensch and offer an explanation to your conversational partner, so they're not startled or embarrassed.
Your religious beliefs are beautiful, compelling and inspiring. Keep them that way by treating others with kindness and respect while you conscientiously put them into action. After all, kindness and respect for you was why you got offered that handshake - be nice when you turn it down.
Yesterday, at a Shiva in a reformed Jewish household that does not observe this rule, a male mourner silently refused to shake my hand with no explanation. Ah, I realized, he cannot shake my hand. I saw that right away, and withdrew my hand. I must admit, though, that I felt awfully sorry for his sister who watched his silent rebuff -- I was fine, but she was embarrassed at his rudeness toward her Shiva caller. Poor guy.
Instead of allowing his religious beliefs to improve him (by kindly explaining) and inform me (by opening my eyes to his belief), the young man diminished us all by acted rudely. My advice: Don't be rude. Speak up: "I can't shake your hand" Show you're a mensch: "It's nice to meet you."