The NUnit 3.0 driver cannot support this test assembly

This is intended to be a self-answered question, but if someone can provide a better answer, please do!

I'm trying to get started with NUnit. I installed NUnit 3.4 using the MSI installer, and I also installed the 3.2 test adapter. Then I created a simple project and a separate classes library for unit tests. Then I added a reference to the NUnit framework. The library builds fine, but when trying to run it either through the test adapter or using the command line runner, I get this message:

The NUnit 3.0 driver cannot support this test assembly. Use a platform specific runner.

There are no specific details as to why it cannot support this particular harmless test assembly. The test adapter provides an exception name as well, but it isn't particularly useful either.

The question is: what's going on and, most importantly, how to fix it? The answer is coming up in a minute.