We have analyzed IRAS image data using a random position, multiple- aperture photometry method to study diffuse far-infrared emission for a sample of 56 clusters of galaxies at 60 and 100 microns. Five of the clusters in the sample exhibit excess far-infrared emission, but at low levels of significance. For A496 (Perseus) and MKW 1 this emission appears to be associated with a strong point source: in A2344, the observed excess is probably due to a foreground cirrus cloud. Two clusters, A262 and A2670, show evidence for extended diffuse emission. This emission is unlikely to be due to point sources associated with the clusters. In these two clusters, the diffuse emission shows evidence of being extended along the cluster's major axis. Far-infrared maps at 60 and 100 microns are presented for these five clusters. We find that contamination due to diffuse Galactic cirrus represents the major source of uncertainty in the measurement of far-infrared emission. The mean cluster fluxes for the sample as a whole are 44 and 107 mJy for a 4' diameter aperture at 60 and 100 microns, respectively. The corresponding fluxes for a 10' diameter aperture are 138 and 253 mJy at 60 and 100 microns. Photometry for the entire sample shows statistical evidence for excess emission at 60 microns associated with clusters in apertures 4' and 10' in diameter. This result is unlikely to be caused by random sampling errors. The flux distributions are analyzed for evidence that excess far-infrared radiation correlates with cluster environmental characteristics. Although we find evidence that poorer X-ray clusters are stronger fare emitters, this result is tentative owing to the small sample sizes involved. The far-infrared luminosities, dust temperatures, and dust masses which these results imply are also presented. Finally, the origin and possible heating mechanism for dust in the intracluster medium is discussed.