You guessed correctly. You do not file a tax return as the inheritance check (cash) is nontaxable income to you. Don't file this year or next - assuming your only income is SS or inherited money.
A handful of states (Iowa, Kentucky, Maryland, Nebraska, New Jersey, and Pennsylvania) have some kind of inheritance tax. Refer to your state's Department of Revenue to see what the requirements are for filing an inheritance tax.
Interest or other income generated from inherited cash would be taxable. For example, if you inherited $10,000 in cash and deposited it into a bank account, generating $40 of interest this year, the $40 would be included in your income as taxable interest. But if the interest is not enough to create a filing requirement, you do not have to file it.
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