I am in the process of finishing my PhD (in chemistry) in the US. Currently I am applying for post-docs (also in the US) and have gotten positive responses from a number of professors I would be interested in working for. Some of them indicated that it would be necessary for me to find my own funding.
I have a somewhat unusual history; I am a German citizen, did my undergraduate studies in Switzerland and my PhD in the US.As far as I can tell without close ties to the German or European research community I do not qualify for any of the more common German/European fellowships like DAAD or Marie Curie. Humboldt is also not an option, since the professors in question are not Humboldt fellows. American funding through NIH, ACS or NSF also does not seem to be an option for non-Americans.
Are there any other funding agencies I may have missed? Does anybody have experience with a similar situation?

Does it need to be a fellowship or can you just look for a post doc as an employee? I thought that there was no restriction on nationality if you were employed by a university? I know there can be restrictions if you apply to the federal organisations like the USDA. I saw several international students and post docs when I was in the U.S. and I imagine they were just on J1 visas?

thank you for your reply! I think what you mentioned is going to be my only option and I am optimistic that this can work if I can't find another option. It is just unfortunate that in one case the PI I applied to only accepts post-docs with independent funding.