After some additional poking around the UI, I figured out a decent way to do this:
- Open the Test Explorer window. (Test menu > Windows > Test Explorer)
- Click on the little down arrow just to the left of the search field.
- From the context menu that appears, select Project. The Test Explorer window contents change to be grouped by project in a 2-tier hierarchy.

- Right-click on the desired project.
- From the context menu that appears, click Run Selected Tests.

I’d still like a reasonable way to do this with only the keyboard, though!