According to the Shulchan Aruch, we wash our hands right-left-right-left-right-left (with a cli). We wash our hands in the morning for three reasons: B'ria Chadasha (a new creation, because it's like we are reborn when we wake up); Ruach Ra'a (the 'bad spirits', or germs because way back when they didn't understand them (this is why we have to pour three times, to remove this)), and Niki'ut (cleanliness).
But why do we have to wash our hands b'seirugin (right-left-right etc.)?