I don't know what I'd do without Blu Tac! It's just the greatest thing ever.
I also save takeout food containers (the Chinese and Mexican restaurants often use these sturdy ones, below) for everything from part organization to resting the model in progress. The edges tend to be sharp, so I just put some Tamiya tape on them. Add in some toothpicks, blu tac, and a 3 x 5 card and not only is everything easier to hold, but your risk of losing a part to the carpet monster is significantly diminished.