Thanks for all the ideas. Here is what I came up with:
1. I use one plane as the ground, and the other one for power.
2. To connect something e.g. a capacitor, from ground to power, I remove the copper around one hole with a track cutter on the top for one lead, and on the bottom for the other lead.
3. Now I solder the leads, making sure they are short enough not to touch the perimeter of the circle I cut into the copper.
4. For anything other than ground or power I use Manhattan-style glue-on little PCB pieces.
Result: This is actually really cool, and gives really good power/ground planes!!!
Example with a resistor going from power to ground is attached (not that that's useful, just experimenting)
I think I will go back to the flea market next week and see if the guy has any left, I really like the idea of using this for RF stuff.