If you are one of those that just can't resist the urge to use a hole bunch of solder, there may be an alternative method of cleanup. When I worked in electronics we used a product called solder wick to soak up excess solder. Now that was low temperature solder so I had no idea if it would work but thought I would give it a try with silver solder.
First I melted a big glob of solder on a piece of copper sheet, cooled and quenched it. Then I put the copper sheet in a third hand (to act as a heat sink). I put some flux on one end of a one inch piece of copper braid (Rio Grande p/n 132-526 FB 1/8) and held the other end in a pair of cross lock tweezers. Holding the fluxes end of the braid on the solder I put the flame of my torch on just the braid. When the braid got to the melting point of the solder, the solder melted and wicked up into the braid. Be careful not to over heat the fine wires of the braid and melt them into your piece. There will still be a thin film of solder on your piece but that's better than a big old ugly gob.