We are conducting a search for supermassive black holes (SMBHs) with masses below 10^7 M_sun by looking for signs of extremely low-level nuclear activity in nearby galaxies that are not known to be AGNs. Our survey has the following characteristics: (a) X-ray selection using the Chandra X-ray Observatory, since x-rays are a ubiquitous feature of AGNs; (b) Emphasis on late-type spiral and dwarf galaxies, as the galaxies most likely to have low-mass SMBHs; (c) Use of multiwavelength data to verify the source is an AGN; and (d) Use of the highest angular resolution available for observations in x-rays and other bands, to separate nuclear from off-nuclear sources and to minimize contamination by host galaxy light. Here we show the feasibility of this technique to find AGNs by applying it to six nearby, face-on spiral galaxies (NGC 3169, NGC 3184, NGC 4102, NGC 4647, NGC 4713, NGC 5457) for which data already exist in the Chandra archive. All six show nuclear x-ray sources. The data as they exist at present are ambiguous regarding the nature of the nuclear x-ray sources in NGC 4713 and NGC 4647. We conclude, in accord with previous studies, that NGC 3169 and NGC 4102 are almost certainly AGNs. Most interestingly, a strong argument can be made that NGC 3184 and NGC 5457, both of type Scd, host AGNs.