Look at your DataSource

dLMessages.DataSource = pds;
dLMessages.DataBind();

where pds is DataView, which means you need to cast DataItem to DataRowView. So you will have

protected void dLMessages_DataBound(object sender, DataListItemEventArgs e)
    {
        DataRowView msgItem = e.Item.DataItem as DataRowView
            if(msgItem != null)
            {
                if(Convert.ToBoolean(msgItem.Row.Item["AcceptedAnswer"].ToString())
                {
                   dLMessages.ItemStyle.BackColor = Color.FromName("#A0FE96");
                }
            }
    }

Item.DataItem represents data for each Item in your DataList, because your dataSource is list of DataRowViews, you need to cast Item.DataItem to DataRowView ( because Item.DataItem return type of object ). Then you need to check AcceptedAnswer on your datasource, to see whethere backcolor should be seted or not.