Observations at 70 microns with the Spitzer Space Telescope have detected several stellar systems within 65 pc of the Sun. Of 18 presumably young systems detected in this study, as many as 15 have 70-micron emission in excess of that expected from their stellar photospheres. Five of the systems with excesses are members of the Tucanae Association. The 70-micron excesses range from a factor of ~2 to nearly 30 times the expected photospheric emission from these stars. In contrast to the 70-micron properties of these systems, there is evidence for an emission excess at 24 microns for only HD 3003, confirming previous results for this star. The lack of a strong 24-micron excess in most of these systems suggests that the circumstellar dust producing the IR excesses is relatively cool (T_dust < 150 K) and that there is little IR-emitting material within the inner few AU of the primary stars. Many of these systems lie close enough to Earth that the distribution of the dust producing the IR excesses might be imaged in scattered light at optical and near-IR wavelengths.