What is considered best practice for using color in website text?

I'm in charge of a website that looks more or less like this:

It needs some work, but I'm mostly worried about defending one aspect of the design: keeping the text black or at least grayscale. My main argument for this is because color is used for links and buttons to make them stand out from text that is not interactive. However, my clients want to "keep the website from looking like a black and white term paper."

What research is there on the effectiveness of keeping body and headings black/grayscale vs. colored to help distinguish them from links? What is generally considered best practice when deciding to make textual elements colored?

This question may generally overlap with design, but to keep it on topic I'm specifically concerned about usability, accessibility, etc.