Is there a better waiting pattern for c#?

A much better way to implement this pattern is to have your Thing object expose an event on which the consumer can wait. For example a ManualResetEvent or AutoResetEvent. This greatly simplifies your consumer code to be the following

if (!Thing.ManualResetEvent.WaitOne(sleep_time)) {
  throw new ItDidntHappen();
}

// It happened

The code on the Thing side is also not really any more complex.

public sealed class Thing {
  public readonly ManualResetEvent ManualResetEvent = new ManualResetEvent(false);

  private void TheAction() {
    ...
    // Done.  Signal the listeners
    ManualResetEvent.Set();
  }
}