We present high resolution (similar to 0.01 cm(-1)) infrared absorption spectra of the nu(4) band of methane doped parahydrogen (CH4/pH(2)) solids produced by two different techniques: gas condensation in an enclosed cell at T approximate to 8 K, and rapid vapor deposition onto a T approximate to 2 K substrate in vacuum. The spectrum of the rapid vapor deposited solid contains a novel progression of single peaks with approximate to 5 cm(-1) spacing, superimposed over the known spectrum of CH4 molecules trapped in sites of D-3h symmetry in hexagonal close-packed (hcp) solid pH(2). New theoretical calculations of the rovibrational transitions of a tetrahedral molecule in an external field of O-h symmetry permit the assignment of this new progression to CH4 molecules trapped in crystalline face centered cubic (fcc) regions of the pH(2) solid. Annealing of the rapid vapor deposited samples to T approximate to 5 K decreases the intensities of the CH4/pH(2)(fcc) absorptions, and results in intensity changes for parallel and perpendicularly polarized CH4/pH(2)(hcp) transitions. We discuss these phenomena, and the narrow (0.01-0.04 cm(-1) full width at half-maximum) absorption linewidths, in terms of the microscopic structure of the pH(2) hosts.