The easiest way to get to the bulbs is to remove the cluster, which only takes about 10 minutes. Instead of using the hooks, remove the panel above the drivers knees and reach up behind the dash to push it out. There is nothing holding it in place except wires, an oil line, and the speedo cable. The bulbs are actually located at the back of the cluster with a "fiber optics" piece of plastic that transports the light to the front of the cluster. You may be able to pull out the bulbs without disconnecting the wires, oil line, or cable (depending how much slack you have in the cable) by just pulling the cluster partially out. It's fairly obvious once you see it. The bulbs are just standard little instrument bulbs that you can get at a parts store. If replacing the bulbs does not work, it's probably the dimmer. You will have to remove the cluster to either replace of bypass the dimmer if that's the problem.

Edit: If the problem is the dimmer, it would also affect the other dash lights on the heater controls and the switches.