There is a three-year exemption from Greek taxes for American professors teaching in Greece. After three years, presumably the American professor will be liable for Greek taxes. My question is: Is there a way to restart the three-year exemption period by leaving Greece for a certain period and then returning, or is the three-year exemption only a one-time exemption?

Normally exemption is one-time.

Thank you. When you say "normally," I assume you are relying on your general knowledge of tax exemptions, and not this one in particular. Is that correct?

This forum deals only (or to 99.999%) with US taxes, so I believe Mukat's answer is a general answer and does not need to apply to your specific question (but most likely does). You might want to ask a tax professional familiar with Greek tax law, the American embassy in Greece should be able to give you some names.

You probably knew that already, but you still need to file tax returns in the US. Most likely you will qualify for the foreign earned income exclusion, but you are still required to file.

Having reviewed the U.S.-Greece Tax Treaty, I find no provision that addresses extensions of the three-year exemption period, or whether the exemption is consiered to be one time or not.

Given that fact, it is likely that the Greek authorities will only grant the tax exemption ONCE in a lifetime, especially given the dire financial straits Greece is in right now.

The tax treaties are reciprocal. Here is from IRS pub 910. Tax Treaties.
"A professor or teacher who is a resident of Greece and who is temporarily in the United States to teach at a university, college, or other educational institution for a maximum of 3 years is exempt from U.S. income tax on the income received for teaching during that period."
Yes, I did not give 100% sure answer. Also I am not aware of treaty with any country where such exemption can be taken twice.

Actually, there ARE treaties which specifically allow the exemption to be used more than once.

But the default position would be to NOT allow it a second time unless the treaty allows it.