There is absolutely no requirement for overseas voters to continue to maintain a residence or to own property in the U.S. in order to vote.
U.S. citizens living outside of the U.S. register and vote in the state and county where they last established residence (domicile) in the U.S. before moving outside of the country, this includes voters that no longer maintain U.S. residence.
You will use the address of the last real home you had in the U.S.—where you actually resided. This is your "voting residence address", and it is this address that defines your state and jurisdiction for voting. You cannot use a P.O. Box as your last U.S. address.
If you are a U.S. citizen but have never lived in the U.S., many states allow you to vote. Your voting residence address would be your U.S. parents’ last residence address. If you are unsure of what address to use, you can see the question “Which U.S. address do I use when registering to vote from overseas?” in the Address-related section of this FAQ.
Note that if you intend to live outside the United States for the long term and you aren't maintaining your domicile in your previous state of residence, under federal law you maintain the right to vote for federal offices (President, Vice President, U.S. Senator, and U.S. Representative). That federal law is called The Uniformed and Overseas Citizens Absentee Voting Act (UOCAVA).
To be eligible to vote for state and local offices (for offices such as Governor, State Legislator, County Clerk, Mayor, etc.), typically the voter must maintain residence/domicile in his/her last state of residence even when living overseas. For more details, you will need to check the laws of your state, which you can do so by going to the State Voting Requirements page, inputting your state, and looking under the “Eligibility Requirements” tab.
If you are serving in the uniformed services, or are the spouse or dependent of someone serving in the military, you can find information under the question "As a uniformed services member, spouse or dependent, how do I determine my legal residence address for voting purposes?" in the Address-related section of this FAQ.