I have two separate circuit boards (clock & radio) that I'd like to place within the same enclosure. One takes 12V, and the other needs 3V. How can I utilize one AC adapter to power both boards appropriately?
Use a 12V DC output AC Adapter to power the 12V module directly. You will then need a voltage regulator or DC to DC converter to reduce the 12V down to the 3V required for your other module. If the current requirement for the 3V module is low you could build a linear regulator out of an LM317, two resistors, two capacitors and a heatsink.
If your current requirement is more than a small number of mA the LM317 style design will get very hot and so a switching type DC-DC converter module would be required. You can find these for sale on eBay. Building your own switching regulator requires a certain amount of experience with such designs. Here is a sample type thing from eBay that could suit your needs.
(image from above linked eBay auction page)
Here is a diagram of how the power lines from the 12V wall adapter would connect into your two modules.
Based on the structure of your question, I am assuming that the AC adapter will supply the necessary 12V... For the 3V, you will will need to get a voltage regulator. This is very simple to make: typical linear regulators are cheap, three prong components that resemble transistors. You will most likely need a leveling capacitor on the input and output terminals of the regulator, but the datasheet should help you make that decision.
Although, 3.3V is a lot more common than 3V. You should check if your board can tolerate 3.3V instead of 3V. That will make it a lot easier. A popular choice for this application would be the LM317 variable regulator.
If the circuit requires a decent amount of current, it may be more advisable to use a switching regulator instead for higher efficiency.