I have a masters in this stuff

Research, as usual, is very mixed, but it seems that there are a few things that seem to really stick out:
1. Most of the time, an effective weight training program improves performance in any sport, including cycling (that takes us to #2).
2. Most of the available research that says weight training doesn't work for cycling has a severe flaws in their training programs (poor program design, no progression, too short, unsupervised, etc.)
3. Choice of subjects has an effect on the outcome: The biggest improvements are seen in people who are either novices and/or of below average strength. A research study using highly trained subjects may not see statistically significant differences in cycling ability. (caveat- as we all know, sometimes the difference between 1st & 2nd place isn't statistically significant, either!)

So, what does this mean?
1. You should lift weights
2. You need an effective training program. My advice would be to talk to someone with a CSCS (certified strength and conditioning specialist) certification. NOT your run-of-the-mill personal trainer, and NOT a cycling coach (even if they're awesome for training you on your bike, you need to talk to someone who is trained in strength and conditioning for sport). A CSCS will prescribe the right amounts of strength and power exercises at the right times in order to maximize your performance.
3. If you are generally a bit weak or haven't been training for long, you're going to see bigger improvements than if you are a very seasoned rider (see caveat above, though).

Weight training improves cycling (and running) in a couple of ways. The obvious way is by increasing the ability to produce force. This will have the greatest impact on things like sprinting and climbing. Almost more importantly, strength training makes your movements more efficient. You make the muscle stronger, so each pedal stroke (or stride) requires less physiological effort from your body, conserving energy.

Hopefully this helps. I didn't read the "spirited" discussion posted above because I have probably had the same one with some of my cycling buddies