Traveling on a TN Visa: Multiple Entry

You may exit and re-enter the U.S on a TN visa given that the visa has not expired, and you are currently working for the same employer. Sometimes they will check if you are with the same company as the one that issued you the letter.
See here for more information:
http://www.tnvisabulletin.com/nafta-tn-blog/2010/5/3/vacations-while-on-a-tntd-visa.html

H-3 Visa, an Alternative to H1-B Visa

The H1-B visa is even harder to obtain, now that less than 30% of applicants are lucky enough to win the lottery.

There is an alternative, the H-3 visa. It is a practical training visa for special cases. See below for more information:

https://www.uscis.gov/working-united-states/temporary-workers/h-3-nonimmigrant-trainee/h-3-nonimmigrant-trainee-or-special-education-exchange-visitor

Traveling While Applying for the OPT

You can travel while your OPT application is being processed, however, there are risks involved. I was traveling abroad after I applied for my OPT, but before I received it in the mail. My family is living in the U.S, so I had a stable address that would be able to receive the card. I re-entered on an expired F-1, but I had my application and OPT receipt with me. I had to speak to one of the border officers in the little rooms that are located in the back of the airport, but after I showed him my job offer letter and OPT application I was able to go. I got the card a few days later in the mail.

There are risks involved with traveling before obtaining your card, but as long as you bring proof that you will be starting a job soon, and that you did apply for your OPT card, you should be allowed back into the U.S to wait for it. Otherwise, you would have to get the card shipped to whatever location you are abroad.

Make sure to have all your documents with you, and that you are covered (a.k.a have a job). I would not go abroad if you are unemployed and waiting for your OPT card.