Orthodox people kiss everything: icons, relics, hands, cheeks, it doesn’t seem to matter, and few things are safe from an Orthodox person planting a smooch on it! We kiss hands of clergy, out of reverence for hands that have held God. We kiss icons (on the hands or feet, avoid kissing the face) to give honor to that holy person. We kiss each other because we want to foster love and friendship among the people of God.
Often, clergy and laity alike will come in for a kiss when you least expect it. Careful! Don’t kiss your priest on the lips (unless you’re married to him), and be prepared for returns: Greeks will usually want to kiss you once on each cheek, and Russians will go back for a third kiss. If you’re kissing cheeks, start with the right cheeks, then go for the left (try not to bump noses!), and brace yourself for one more on the right cheek if you’re in a Slavic community! It is not uncommon for such kisses to be imaginary kisses, where you basically just touch cheeks and make a smooching sound in the air. Depends on who is doing it, and likely has something to do with how afraid they are of your face hosting germs!
In the middle of the service, there is a “passing of the peace”—the priest says, “The peace of the Lord be always with you.” In recent centuries, the traditional Orthodox practice consisted of clergy around the altar kissing each other, but some parishes essentially have a brief meet-and-greet in the service. Visitors can probably get away with handshakes to those around them, but stick around long enough and people might start kissing you. In some parishes, one of several customary greeting are exchanged at this time, too.
Sidebar: In many Slavic parishes, it is customary for communicants to kiss the communion chalice after partaking of the Holy Eucharist. This is done out of great reverence for the chalice that holds the Author of Life. In most Byzantine parishes, however, they wouldn’t dream of doing this, also out of great reverence for the chalice that holds the Author of Life. It’s just one of those quirky differences in Orthodoxy!