We have observed the HCO+ J=3-2 line toward 17 starless cores selected from the list of Ward-Thompson and coworkers. Six of these cores have line asymmetries indicative of collapse. The excess of blue-skewed profiles over red-skewed profiles is at least as large as that found in samples of Class 0 and early Class I sources. The observed line profiles have the same narrow line widths and small peak temperatures predicted for young sources in evolutionary models, but the blue/red ratios, like those of older sources, are higher than models predict. The infall signature also occurs over large scales, suggesting that these cores have overall inward motions. We have divided these starless cores into two groups based on the continuum photometry of Ward-Thompson and coworkers and our HCO+ data. We find stronger HCO+ emission among the cores detected in the submillimeter, and all the blue-skewed line profiles are in this group, supporting the suggestion of Ward-Thompson and coworkers that these are the pre-protostellar cores.