I've had trouble finding much documentation on using LinkedHashMap with a ListView for Android, but I was able to get the list to populate correctly with the below.
Unfortunately, when I scroll, the list is scrambled the same as it would be if I was using a regular HashMap. I'm not sure what I'm doing wrong.
MainActivity.java (only a portion):
public class MainActivity extends ListActivity {
    private ListView dataList;
    ArrayList<String> listItems=new ArrayList<String>();
    ListViewAdapter adapter;
    private ArrayList<LinkedHashMap<String, String>> list;
    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_main);
        dataList = (ListView)findViewById(android.R.id.list);
        list=new ArrayList<LinkedHashMap<String,String>>();
        adapter=new ListViewAdapter(this, list);
        dataList.setAdapter(adapter);
        addItems();
    }
    public void addItems() {
        LinkedHashMap<String,String> row = new LinkedHashMap<String, String>();
        row.put(KEY_COLUMN, key);
        row.put(FIRST_COLUMN, dateTime);
        row.put(SECOND_COLUMN, data1);
        row.put(THIRD_COLUMN, data2);
        row.put(FOURTH_COLUMN, data3);
        list.add(row);
        adapter.notifyDataSetChanged();
    }
}
ListViewAdapter Class
public class ListViewAdapter extends BaseAdapter{
    public ArrayList<LinkedHashMap<String, String>> list;
    Activity activity;
    TextView txtKey;
    TextView txtFirst;
    TextView txtSecond;
    TextView txtThird;
    TextView txtFourth;
    public ListViewAdapter(Activity activity,ArrayList<LinkedHashMap<String, String>> list){
        super();
        this.activity = activity;
        this.list = list;
    }
    @Override
    public int getCount() {
        // TODO Auto-generated method stub
        return list.size();
    }
    @Override
    public Object getItem(int position) {
        // TODO Auto-generated method stub
        return list.get(position);
    }
    @Override
    public long getItemId(int position) {
        // TODO Auto-generated method stub
        return 0;
    }
    @Override
    public View getView(int position, View convertView, ViewGroup parent) {
        // TODO Auto-generated method stub
        if(convertView == null){
            LayoutInflater inflater = activity.getLayoutInflater();
            convertView = inflater.inflate(R.layout.list_row_columns, null);
            txtKey = (TextView) convertView.findViewById(R.id.row_key);
            txtFirst = (TextView) convertView.findViewById(R.id.date_time);
            txtSecond = (TextView) convertView.findViewById(R.id.data1);
            txtThird = (TextView) convertView.findViewById(R.id.data2);
            txtFourth = (TextView) convertView.findViewById(R.id.data3);
        }
        LinkedHashMap<String, String> map = list.get(position);
        txtKey.setText(map.get(KEY_COLUMN));
        txtFirst.setText(map.get(FIRST_COLUMN));
        txtSecond.setText(map.get(SECOND_COLUMN));
        txtThird.setText(map.get(THIRD_COLUMN));
        txtFourth.setText(map.get(FOURTH_COLUMN));
        return convertView;
    }
}
 
    