You can use XmlWriterSettings and set the property OmitXmlDeclaration to true as described in the msdn. Then use the XmlSerializer.Serialize(xmlWriter, objectToSerialize)as described here.
To put this all together - this works perfectly for me:
// To Clean XML
public string SerializeToString<T>(T value)
{
var emptyNamespaces = new XmlSerializerNamespaces(new[] { XmlQualifiedName.Empty });
var serializer = new XmlSerializer(value.GetType());
var settings = new XmlWriterSettings();
settings.Indent = true;
settings.OmitXmlDeclaration = true;
using (var stream = new StringWriter())
using (var writer = XmlWriter.Create(stream, settings))
{
serializer.Serialize(writer, value, emptyNamespaces);
return stream.ToString();
}
}