The Descendents are XElement instances, not strings, so you need to Select the Value property in order to get strings:
public static List<string> checksum(string cusip)
{
XDocument doc = XDocument.Load("SampleXML.xml");
List<string> aacruedlist = doc.Descendants("PositionSummary")
.Where(item =>
{
string cus = (string)item.Element("Cusip");
return cus != null && cus == cusip;
})
.Descendants("AccruedInterest")
.Select(d => d.Value)
.ToList();
return aacruedlist;
}