I started working full time while still holding F1 student visa. At that time, I was exempt from paying Social security and Medicare. A year later, my status was changed to H3B, which is a trainee visa (similiar to H1B except it allows a person to work only for the company that sponsored you). My company did not change my visa status respectively and as the result none of SS or Medicare was withheld from my 2008 pay.
Please confirm that as a H3B visa holder I need to pay SS and Medicare.

Yes, the "H" visa is a work visa.

You ARE liable for Social Security and Medicare taxes.

Notify your employer of that fact, preferably in writing (email is good for that).