I received an inheritance from a non-resident alien relative who lived in a foreign country. Do I have to report this foreign money even though an inheritance in the United States is not considered "income"?

Only if the total amount you receive is valued at more than $100,000. If so, you must complete Form 3520Annual Return to Report Transactions with Foreign Trusts and Receipt of Certain Foreign Gifts. This is separate from your income tax return filing requirement and generally will not result in taxable income provided the money is a legitimate gift or bequest.