I figured it out. Here's a class, compiled using JDK 6, that works.

import java.util.logging.*;
import java.util.Date;
import java.io.*;
import java.lang.*;

class IndependentLog implements Runnable {
   Logger myLogger;
   IndependentLog(Logger myLogger) {
    this.myLogger = myLogger;
   }

   private void logOutSideRunMethod() {
    myLogger.finest("It doesn't get any more detailed than this.");
   }

   public void run () {
    myLogger.severe("Hello from a different thread!");
    logOutSideRunMethod();
   }
}

/**
* Example of manipulating a logger.
*
* Send the log messages to a file, but not
* to the console.  Format the log messages
* to several lines, using class LogRecord's
* "get" methods to indicate all the good
* things you can log, including which thread
* you're in.
*
* Chris Williamson, April 3, 2008
**/
public class LogExample  {
   Logger myLogger;
   String logFileName = "c:\\LogExample.log";
   
   public void sillyMethod() {
    myLogger.warning("Don't mess with silly methods!");
   }
   public static void main(String [] args) {
       
    LogExample le = new LogExample();
    le.myLogger = Logger.getLogger("myLogger");
    FileHandler myHandler = null;

    System.out.println("Look in file " +
              le.logFileName +
              " to see log records.");

    le.myLogger.setLevel(Level.ALL);

    /* turn off logging to the console. */
    le.myLogger.setUseParentHandlers(false);

    /* Send the logging messages to a
    * file. Will overwrite the file
    * every time it's run. */
    try {
       myHandler = new FileHandler(le.logFileName);
    }
    catch(IOException e) {
       System.out.println("Can't open file handler file. exiting.");
       System.exit(1);
    }

    Formatter myFormatter = new MyFormatter();
               
    le.myLogger.addHandler(myHandler);
       
    myHandler.setFormatter(myFormatter);
    le.sillyMethod();
    le.myLogger.severe("Things are terrible here!");
    le.myLogger.info("Just letting you know.");

    /* Set up a separate thread and send
    * a log message from there. */
    IndependentLog il = new IndependentLog(le.myLogger);
    Thread nt = new Thread(il, "New Thread!");
    nt.start();
   }
}

/**
* From http://tinyurl.com/26pybz with changes.
*
*/
class MyFormatter extends Formatter {
   public MyFormatter() {
    super();
   }
   public String format(LogRecord record) {
       
    // Create a StringBuffer to contain the formatted record
    // start with the date.
    StringBuffer sb = new StringBuffer();

    sb.append("\n\n**NEW LOG RECORD**\n");

    sb.append("Sequence Number: " +
         record.getSequenceNumber() +
         "\n");
    Date date = new Date(record.getMillis());
    sb.append("Date: " +
         date.toString());
    sb.append("\n");
    sb.append("Thread Name: " +
         Thread.currentThread().getName() +
         "\n");
    sb.append("Thread ID: " +
         record.getThreadID() +
         "\n");
    sb.append("Class name: " +
         record.getSourceClassName() +
         "\n");
    sb.append("Method Name: " +
         record.getSourceMethodName() +
         "\n");
    // Get the level name and add it to the buffer
    sb.append("Level: " +
         record.getLevel().getName() +
         "\n");
       
    // Get the formatted message (includes localization and
    // substitution of parameters) and add it to the buffer
    sb.append("Raw Message: " +
         formatMessage(record) +
         "\n");
   
    return sb.toString();
   }
}