Well you have a few options. First, if you put the files in the trash, just plugging the thumbdrive in and emptying the Mac trash should clear it. Second, you could reformat the thumdrive by Going to Applications>Utilities>Disc Utility, opening it, selecting the thumbdrive from the list on the left, selecting the 'erase' tab and erasing the drive to FAT32 format, but this will clear ALL data on the drive. Might be the easiest though.
Finally, you can do all this using the terminal, but if you don't have experience its a little intimidating, and has some potential to go wrong. Personally, I recommend any OSX user with an interest in the guts of their machine to go and find a tutorial for the Terminal as its just useful.