Studies say that modern humans didn’t interbreed with Neanderthal becaue N. genes aren’t present in our genetic material. How do they know what a Neanderthal gene looks like? Besides, I’ve seen guys on the subway that look…but I digress.
They extracted the DNA from Neanderthal bones. This article gives some details on recent work.
Note that the analysis so far (AFAIK) has been exclusively on Neanderthal mitochondrial DNA (from outside the cell nucleus). There remains a small possibility that analysis of nuclear genes could show some relationship between Neanderthal and modern populations.
Specifically, if Neanderthal men were mating with modern-type women, the decendents would show no trace of the Neanderthal’s mitochondrial DNA. Whether this was the case, I cannot say.