There are two common stories behind the use of the phrase "getting paid under the table". It is impossible to know for certain which one is the actual originator so I will include both.
This first is; before paper money gold itself was used as currency. Most people would have gold coins, bars and jewelry and when they would purchase something they would simply cut a sliver off and use that as payment. When the boss would gather everyone up to distribute pay he would usually sit at his dining table and make piles for each person often cutting a bit of gold from a larger chunk. While gold is pretty soft this would leave a great deal of wear and gouges in the table top. Tables were often reversible. You would flip the top over so that one side was nice for eating, company and everyday use and the other or the "under side" of the table would be flipped up when it was time to pay the staff.
The other is somewhat more straight forward. The belief is that it stems from the act of passing a bribe in an envelop under the table to an official or whoever you were trying to pay off.