MimeMultipart content = new MimeMultipart("alternative");
MimeBodyPart text = new MimeBodyPart();
MimeBodyPart html = new MimeBodyPart();
          
text.setText( oMail.getStrBody() );
text.setHeader("MIME-Version" , "1.0" );
text.setHeader("Content-Type" , text.getContentType() );
          
html.setContent(oMail.getStrHTMLBody(), "text/html");
html.setHeader("MIME-Version" , "1.0" );
html.setHeader("Content-Type" , html.getContentType() );

content.addBodyPart(text);
content.addBodyPart(html);
          
message.setContent( content );
message.setHeader("MIME-Version" , "1.0" );
message.setHeader("Content-Type" , content.getContentType() );
message.setHeader("X-Mailer", "Recommend-It Mailer V2.03c02");
message.setSentDate(new Date());

I found that the above was useless for me as well. Here is what I needed to do to send a multi-part text/html email. You may need to edit the line:
props.put("mail.smtp.host", "localhost");

You will need to edit the from and to emails.

// Create the message to send
   Properties props = new Properties();
   props.put("mail.smtp.host", "localhost");
   Session session = Session.getInstance(props,null);
   MimeMessage message = new MimeMessage(session);

// Create the email addresses involved
   InternetAddress from = new InternetAddress("from@from.com");
   InternetAddress to = new InternetAddress("to@to.com");

// Fill in header
   message.setSubject("I am a multipart text/html email" );
   message.setFrom(from);
   message.addRecipient(Message.RecipientType.TO, to);

// Create a multi-part to combine the parts
   Multipart multipart = new MimeMultipart();

// Create your text message part
   BodyPart messageBodyPart = new MimeBodyPart();
   messageBodyPart.setText("Here is your plain text message");

// Add the text part to the multipart
   multipart.addBodyPart(messageBodyPart);

// Create the html part
   messageBodyPart = new MimeBodyPart();
   String htmlText = "<H1>I am the html part</H1>";
   messageBodyPart.setContent(htmlText, "text/html");

// Add html part to multi part
   multipart.addBodyPart(messageBodyPart);

// Associate multi-part with message
   message.setContent(multipart);

// Send message
   Transport.send(message);

      MimeMessage message = new MimeMessage(session);

      // Create a multi-part to combine the parts
      MimeMultipart multipart = new MimeMultipart("alternative");

      // Create text message part
      MimeBodyPart messageBodyPart = new MimeBodyPart();
      messageBodyPart.setText(textBody);
      multipart.addBodyPart(messageBodyPart);

      // Create html part
      messageBodyPart = new MimeBodyPart();
      messageBodyPart.setContent(htmlBody, "text/html");
      multipart.addBodyPart(messageBodyPart);

      // Associate multi-part with message
      message.setContent(multipart);
      message.saveChanges();

The above works great for browsers/servers that support html content. Internet services that don't support the used character set give the following error. I've seen postings mentioning this error, and have tried to follow the suggestions. Not sure what I am missing in the code above. Also, if I change "alternative" to "mixed", both (html and text) browsers receive text email with html as an attachment (as pointed out in an earlier message in this thread). Thanks for any help that you can provide.