Ever hear about this? Of course. The oil drip can be caused by spillage onto your engine or other car components when the mechanic changed your oil. That is usually the reason, and is the result of sloppy work. However, all it would take to solve that problem is to wipe the spilled oil off of your car components. Another, more serious, reason is that the oil filter was not properly installed. It may have been cross-threaded when the mechanic tried to install it. When this occurs, the oil filter can come completely loose from the engine, causing all of your oil to pour out, and could ruin your engine - which will seize up if it runs very long (just a minute or two) without oil.
On the other hand, the oil drip could be from some other source, such as your transmission. However, the fact that you had no oil drip UNTIL you had your oil changed makes me doubt this.
In any case, have the cause checked out by someone you can trust. Don't take the word of the mechanic who changed your oil.