It depends on whether the state you lived in before you moved requires you to file a tax return.
Some states don't have state income taxes, so if you moved from Alaska, Florida, Nevada, South Dakota, Texas, Washington or Wyoming you aren't required to file a tax return. Tennessee only taxes investment income.
For other states, you may be required to file a "part-year resident" state tax return if you had income while you lived there and it was high enough to require a filing. Also, keep in mind that if you had state tax withheld in a state, you may be able to get a refund of that withholding even if you don't owe taxes and aren't required to file a tax return.
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