I never shake hands, even when I'm not eating. If the person to whom I'm being introduced acts weird because I don't stretch out my hand to meet his or hers, I quickly say "Excuse me, but I have a sprained wrist that's still healing."
However, I've discovered most 'Mericuhns are more intelligent than I normally expect and are just as happy or even more honored if I smile and slightly bow in their direction; especially if I'm eating and they are rude enough to interrupt my meal with friends. If I stand up and give them a slight bow, everyone who expected to shake my hand seems relieved to just accept that I've politely acknowledged their existence in a friendly and open manner.
In Latin countries sometimes we take what seems to be hours greeting everyone in a group by acting like we're kissing each other on the cheek. In reality, we don't really kiss (unless it's immediate family). We just act like we're going to press our cheeks together (but don't) and say whatever the standard greeting is in the local language. This can be especially annoying when you're standing in line to buy stamps or pay an electric bill and one of the employees behind the counter is leaving for the day and has to do the faux cheek press with EVERYONE (even if you don't like most of your coworkers). Everyone stops what they are doing (counting change, stamping your personal check, etc.) and they all do the fake cheek kiss. Latins tend to shake hands only after a group of them, especially business men, have agreed to something (like meet again and continue the conversation). And they all find time afterwards to go to the rest room and wash their hands. Note: they may not wash their hands after taking a piss or shit, but most mens and womens rooms have an attendant on duty who makes certain you receive a paper towel which sort of forces you to wash your hands.
I really dislike shaking hands. In the USA it seems it becomes a contest among men to see who can cause the most pain with the strongest hand shake. Plus, as others have mentioned, you don't know where those hands have been. Remember, approximately 80% of the US population has pin worms. So, next time you catch your boss scratching his ass (or picking his nose), reconsider the idea that NOT shaking hands is rude.