We show specific heat data for Na4C60 and Li4C60 in the range 0.4-350 K for samples characterized by Raman spectroscopy and X-ray diffraction. At high temperatures, the two different polymer structures have very similar specific heats both in absolute values and in general trend. The specific heat data are compared with data for undoped polymeric and pristine C60. At high temperatures, a difference in specific heat between the intercalated and undoped C60 polymers of 100 J K(-1) mol(-1) is observed, in agreement with the Dulong-Petit law. At low temperatures, the specific heat data for Li4C60 and Na4C60 are modified by the stiffening of vibrational and librational molecular motion induced by the polymer bonds. The covalent twin bonds in Li4C60 affect these motions to a somewhat higher degree than the single intermolecular bonds in Na4C60. Below 1 K, the specific heats of both materials become linear in temperature, as expected from the effective dimensionality of the structure. The contribution to the total specific heat from the inserted metal ions can be well described by Einstein functions with TE = 386 K for Li4C60 and TE = 120 K for Na4C60, but for both materials we also observe a Schottky-type contribution corresponding to a first approximation to a two-level system with ΔE = 9.3 meV for Li4C60 and 3.1 meV for Na4C60, probably associated with jumps between closely spaced energy levels inside "octahedral-type" ionic sites. Static magnetic fields up to 9 T had very small effects on the specific heat below 10 K.