I resolved this issue as I was typing the question. I figured I'd answer my question and leave it here for anyone else who may face this issue when using Xcode 6 beta 4.
To resolve this issue, you need to select each of your custom class objects in Storyboard (this includes any custom views, even the custom view controllers themselves).
Then with those objects selected, open the identity inspector and under "Custom Class" you should see the Module option. Click inside the Module text box, and press enter.
That's it! The current module for all of my custom objects must have been internally incorrectly set somehow in Xcode 6 beta 4. But there was no visual indication of this in the inspector.
Note that if pressing enter inside the Module text box doesn't work, try selecting the arrow to the right and manually select your current module, then clear the text box and press enter. You can also try pressing enter inside the class text box (although this usually is to resolve a different issue).
Here is an image to make things more clear: