See: http://en.wikipedia.org/wiki/Galvanized_Yankees

The 3rd Tennessee Mounted Infantry, USA, was a three-months organization, “and did no service, never being fully organized.”
Tennesseans in the Civil War, Part 1, p. 357

What was your grandfather’s name? He should have records in both units.

The 3rd Maryland Battery served in Col. A. W. Reynolds Brigade, MGen. C. L. Stevenson’s Division at Vicksburg.