Nobody has mentioned Kasikornbank yet. They used to be called Thai Farmers Bank. They didn't ask for a work permit when I opened my savings account but this was quite a few years ago (2001).
I went to the branch on Rama 4 Road close to (but not inside) the Tesco Lotus and Carrefour superstores.
Btw, if you need to change money - either way, (changing Thai baht into hard currency is a pain in the neck for example - most banks will simply turn you down, lots of paperwork too) hassle free and for a slightly better rate as well, here's a tip: Go to the World Trade Center (now called Central Plaza or something). Cross the street to where the huge Big C is. Walk past it toward Pratunam/Phetburi road. Just past the second footbridge turn right. Take the next left (there should be a Kasikornbank ahead on your right). Just before you reach the bank turn left. The corner store on your right is called "Superrich" (very subtle, I know).
It will be a little intimidating at first. Go to the counter and tell them exactly what you want. If you can speak at least a modicum of Thai (especially the numbers, you know, nueng roi (=100), song roi, sarm roi, si roi, ha (go up while saying it) roi, hok roi, jet roi, paed (say it slowly) roi, gau roi, nueng pan (=1000), song pan, and so on...) it will be helpful (or take someone you speaks Thai). You will have to wait a few minutes while they put your money together. They will probably call your order number in Thai - or the amount you exchanged, and you won't understand
, but that's ok, it'll be sorted out eventually. You will not get cheated here but count the money anyway. You can check the rate before at the Kasikorn branch. Superrich has always beaten it.
I hope this helped.
Guido