I am an American in Sweden and need to obtain a Chinese work visa. I have a letter.

Do you get your Work Permit?
If you have already got your work permit and the invitation letter, you can try to apply in Sweden. Otherwise, it is hard to tell and you may need to consult the local embassy for helping.

Thanks for your quick response! I only have the letter, not the work Z. Another option is to expedite my passport to Washington, D.C. To have it done by a third party but should I be concerned about shipping my passport overseas, as there's a small chance. For it being lost?

Thanks for your quick response!! Yes, I have a letter from a factory I do business with and I'm only visiting for one week! Any more thoughts??

You will go to that company for a job or just for trading?
If you just go there for trading, the invitation letter from your company is enough for applying and you can applying for a M type. For trading issue, you do not need to apply a work Z. You can apply for it in Sweden if you show them your legal status in Sweden.

ok, I am now in Bangladesh for business so I am hoping the Chinese embassy here in Dhaka will grant me one. Do you think they will?

BTW, Sweden wouldn't issue me that...I'm hoping the office here is more liberal.

It's hard to tell. Normally, it's easier to apply from Chinese embassy in your own country. But you can give it a try. Good luck!