Actually, I am asking for my wife.

She is a German Citizen and a Green Card Holder living here in the US with me (a US citizen) and working on her research. She gets paid from a German Postdoctoral Research Fund. But makes no US-based income whatsoever.

Some have told us that she'd have to be visiting on a J-1 visa to avoid paying taxes and that that would only work for 2 years (then shed have to backpay)... but that if she has a green card she must pay full income tax on this postdoctoral research fund (which is tax free in germany, by the way).

Sorry I can't be more discriptive but this is not my specialty by far.

Does she have to report/pay taxes on these German funds here in the US?

If postdoctoral research fund is income, it is taxable.
If it is scholarship, grant or gift, it may not be taxable.

If you are a degree candidate and if the financial aid (includes scholarship and fellowship) is for tuition fee, other fees, books, supplies and equipment, then it is not taxable. For a degree candidate aid for boarding and travel are taxable. Scholarship and fellowship grants, Your U.S. Tax Return: The U.S. Income Tax Topics 1

Turns out it seems the consensus states: She does have to pay taxes but filing under "Alternate sources of income" as a non-taxfree scholarship. But that because the work produces no tangible marketable product she does not have to pay the Self-employed aspect of this tax. And the tax is reducible by tracking research expenses.

Your assessment is correct. She would report the income (minus the research expenses) on Line #21 of your joint Form 1040.