I finally decided to spend a little time figuring out what's going on, and it turns out, it all comes down to temperature (at least it did for me).
There are quite a few reasons why dough might not rise (dead yeast, other ingredients killing or impeding the yeast's ability to work properly, etc.), but temperature is probably the most common issue.
I knew temperature mattered, but what I didn't realize was just how small the window for error is.
I run hot, so I tend to keep my house pretty cool, especially in the winter (when baking is more fun). I also consider myself way more of a "cook" than a "baker." As such, I'm used to winging it, shooting from the hip, improvising, experimenting, and being altogether way more reckless than is acceptable with baking, where apparently you have to stick to the rules a little (or a lot) more.
I've got an insanely delicious Oatmeal Bread recipe from my mom/aunt that reminds me of childhood and all things good in this world. The recipe says you should put a package of yeast in 3/4 cups of lukewarm water with 1/16 teaspoon of sugar and 1/16 teaspoon of ginger. There are two problems with this. 1. "Lukewarm" can be quite different from one person to another. 2. Ginger is actually an anti-fungal, and yeast belongs to the fungus family!
I wonder if the pinch of ginger is to keep the dough from going crazy. A kitchen that is in constant use is warm or even hot, and my mom and aunt's kitchens were certainly that way. So maybe they needed something to actually keep the yeast in check. My 64 degree kitchen doesn't need any help. It will keep yeast from "doing its thing" on its own.
The other thing I have to worry about in my cold kitchen is the temperature of the mixing bowl. I didn't think that would make much of a difference (thinking it would only drop the dough's temperature a tiny bit), but I was wrong. From now on I will be warming the bowl with hot water and then keeping it in an oven set for proofing.
The final element of fixing my dough-rise problems is actually using the oven to "proof" (placing the kneaded dough in an oven set at a relatively low temperature specifically for rising). I had been doing this already, but not really paying much attention to getting the oven warm enough or keeping it there.
If you don't have an oven that does well for proofing, there are a couple of tricks you can try. My oven doesn't really start keeping temperature until it's around 125 degrees, which is a bit too high. Therefore, I set the oven as low as I can to where it will kick on and off (you can watch the indicator light or listen for the "click"), and then I just keep the door open a bit. This will cause a greater fluctuation in temperature, so to combat that, I simply placed a couple of my iron skillets on the bottom rack to assist with keeping the temperature constant (the skillets act as heat sinks). I used my thermometer to monitor things, and I'm able to keep the oven interior at about 100 degrees, which seems perfect to me.
Finally, and especially with the door cracked open, you will want to add an element to keep the rising dough's atmosphere moist. A lot of people place a damp towel over the bowl, and while that helps a bit, I have started placing a shallow pan of hot water on the bottom rack where the iron skillets are hanging out. Not only that, but I leave the water there when baking the bread, and I've found it really helps with the consistency and texture of the bread!