Here are two design variations on a traditional voltaic cell. Both designs are solder less and easy to construct from common household items. The flower and lamp are basically giant paper batteries with the wetted paper acting as a salt bridge between the wires.
Step 1: The Frame
I used the cardboard from a box of soda as the base of the lamp and construction paper for the shade. After making 6 pairs of fairly evenly spaced holes in the base I rolled the paper and marked where the holes lined up- in this example they were an inch apart. It's important that the wetted paper containing the wires doesn't touch it's neighbor or the current won't know where to flow. Cut v-shaped pieces out near the bottom where the shade connects to the base to prevent that from happening.
Step 2: The Wiring
Poke rows of 8 holes in each segment and cut 6 pieces each of aluminum and copper wire, about 4 inches long. Thread the wire through the holes, making sure to alternate the type of wire. In this example I used 18 gauge copper and aluminum wire, but you could use zinc instead of aluminum.
Step 3: Attaching the Base
Here is a shot of how the threading looks after you attach it to the base. Bend the wires out slightly to keep them from falling through the holes while you work on connecting it.
Step 4: Connecting the Wires
Here's a closeup of how I've twisted the wires together. Because there isn't solder involved it's important to make sure there is a strong connection between the wires- three twists of the wire worked well for me, and allowed me the turn them underneath themselves to form feet for the lamp. Make sure the copper wire of one segment is connected to the aluminum wire of the neighboring segment and so on until you are left with one copper wire and one aluminum wire. Identify the polarity of your LED, fit the leads through the middle of the base and twist the positive lead around the copper wire and the negative lead around the aluminum wire at least 2 or 3 times.