H.R.5941: To amend title 10, United States Code, to authorize taxpayers to designate a portion of their income tax payments to a National Military Family Relief Fund to be used by the Secretary of Defense to assist the families of members of the Armed Forces who are serving in, or have served in, Iraq or Afghanistan.

Establishes in the Treasury the National Military Family Relief Fund to make grants to Armed Forces members who are serving, or have served, in Iraq or Afghanistan to assist the families of such members.

Amends the Internal Revenue Code to: (1) allow every individual taxpayer to designate $1 or more of any overpayment to be paid to the Fund; and (2) in addition to any payment of income tax liability, make a contribution of an additional amount which shall be paid over to the Fund.