How do I cut out the vein/tendon from a chicken breast tendorloin?
Cooking For Engineers has a pretty good description. He writes it better than I can, so I'll just quote him. I've used the second method he describes, but after watching the video below, the other looks much easier...


  The breast halves should each have a flap of meat called the chicken tenderloin (or chicken tenders or strips). Lifting the tenderloin should reveal a white tendon. I recommend two ways to trim off this tough tendon. The most popular method is to grab the tendon with your fingers and pull on it while scraping with your knife to release the tendon from the breast. Continue to lift and scrape until the tendon is completely removed. Alternatively, you can use a sharp boning knife and slice along both sides of the tendon (without slicing through the tendon). Then lift any part of the tendon that has been separated from the breast and use your boning knife the cut any parts where the tendon has not cleanly separated from the meat.


There's also a yotube video here.