You just need to override onCreateDialog in an Activity.

//In an Activity
private String[] mFileList;
private File mPath = new File(Environment.getExternalStorageDirectory() + "//yourdir//");
private String mChosenFile;
private static final String FTYPE = ".txt";    
private static final int DIALOG_LOAD_FILE = 1000;

private void loadFileList() {
    try {
        mPath.mkdirs();
    }
    catch(SecurityException e) {
        Log.e(TAG, "unable to write on the sd card " + e.toString());
    }
    if(mPath.exists()) {
        FilenameFilter filter = new FilenameFilter() {

            @Override
            public boolean accept(File dir, String filename) {
                File sel = new File(dir, filename);
                return filename.contains(FTYPE) || sel.isDirectory();
            }

        };
        mFileList = mPath.list(filter);
    }
    else {
        mFileList= new String[0];
    }
}

protected Dialog onCreateDialog(int id) {
    Dialog dialog = null;
    AlertDialog.Builder builder = new Builder(this);

    switch(id) {
        case DIALOG_LOAD_FILE:
            builder.setTitle("Choose your file");
            if(mFileList == null) {
                Log.e(TAG, "Showing file picker before loading the file list");
                dialog = builder.create();
                return dialog;
            }
            builder.setItems(mFileList, new DialogInterface.OnClickListener() {
                public void onClick(DialogInterface dialog, int which) {
                    mChosenFile = mFileList[which];
                    //you can do stuff with the file here too
                }
            });
            break;
    }
    dialog = builder.show();
    return dialog;
}

Thanx schwiz for idea! Here is modified solution:

public class FileDialog {
    private static final String PARENT_DIR = "..";
    private final String TAG = getClass().getName();
    private String[] fileList;
    private File currentPath;
    public interface FileSelectedListener {
        void fileSelected(File file);
    }
    public interface DirectorySelectedListener {
        void directorySelected(File directory);
    }
    private ListenerList<FileSelectedListener> fileListenerList = new ListenerList<FileDialog.FileSelectedListener>();
    private ListenerList<DirectorySelectedListener> dirListenerList = new ListenerList<FileDialog.DirectorySelectedListener>();
    private final Activity activity;
    private boolean selectDirectoryOption;
    private String fileEndsWith;    

    /**
    * @param activity 
    * @param initialPath
    */
    public FileDialog(Activity activity, File initialPath) {
        this(activity, initialPath, null);
    }

    public FileDialog(Activity activity, File initialPath, String fileEndsWith) {
        this.activity = activity;
        setFileEndsWith(fileEndsWith);
        if (!initialPath.exists()) initialPath = Environment.getExternalStorageDirectory();
            loadFileList(initialPath);
    }

    /**
    * @return file dialog
    */
    public Dialog createFileDialog() {
        Dialog dialog = null;
        AlertDialog.Builder builder = new AlertDialog.Builder(activity);

        builder.setTitle(currentPath.getPath());
        if (selectDirectoryOption) {
            builder.setPositiveButton("Select directory", new OnClickListener() {
                public void onClick(DialogInterface dialog, int which) {
                    Log.d(TAG, currentPath.getPath());
                    fireDirectorySelectedEvent(currentPath);
                }
            });
        }

        builder.setItems(fileList, new DialogInterface.OnClickListener() {
            public void onClick(DialogInterface dialog, int which) {
                String fileChosen = fileList[which];
                File chosenFile = getChosenFile(fileChosen);
                if (chosenFile.isDirectory()) {
                    loadFileList(chosenFile);
                    dialog.cancel();
                    dialog.dismiss();
                    showDialog();
                } else fireFileSelectedEvent(chosenFile);
            }
        });

        dialog = builder.show();
        return dialog;
    }


    public void addFileListener(FileSelectedListener listener) {
        fileListenerList.add(listener);
    }

    public void removeFileListener(FileSelectedListener listener) {
        fileListenerList.remove(listener);
    }

    public void setSelectDirectoryOption(boolean selectDirectoryOption) {
        this.selectDirectoryOption = selectDirectoryOption;
    }

    public void addDirectoryListener(DirectorySelectedListener listener) {
        dirListenerList.add(listener);
    }

    public void removeDirectoryListener(DirectorySelectedListener listener) {
        dirListenerList.remove(listener);
    }

    /**
    * Show file dialog
    */
    public void showDialog() {
        createFileDialog().show();
    }

    private void fireFileSelectedEvent(final File file) {
        fileListenerList.fireEvent(new FireHandler<FileDialog.FileSelectedListener>() {
            public void fireEvent(FileSelectedListener listener) {
                listener.fileSelected(file);
            }
        });
    }

    private void fireDirectorySelectedEvent(final File directory) {
        dirListenerList.fireEvent(new FireHandler<FileDialog.DirectorySelectedListener>() {
            public void fireEvent(DirectorySelectedListener listener) {
                listener.directorySelected(directory);
            }
        });
    }

    private void loadFileList(File path) {
        this.currentPath = path;
        List<String> r = new ArrayList<String>();
        if (path.exists()) {
            if (path.getParentFile() != null) r.add(PARENT_DIR);
            FilenameFilter filter = new FilenameFilter() {
                public boolean accept(File dir, String filename) {
                    File sel = new File(dir, filename);
                    if (!sel.canRead()) return false;
                    if (selectDirectoryOption) return sel.isDirectory();
                    else {
                        boolean endsWith = fileEndsWith != null ? filename.toLowerCase().endsWith(fileEndsWith) : true;
                        return endsWith || sel.isDirectory();
                    }
                }
            };
            String[] fileList1 = path.list(filter);
            for (String file : fileList1) {
                r.add(file);
            }
        }
        fileList = (String[]) r.toArray(new String[]{});
    }

    private File getChosenFile(String fileChosen) {
        if (fileChosen.equals(PARENT_DIR)) return currentPath.getParentFile();
        else return new File(currentPath, fileChosen);
    }

    private void setFileEndsWith(String fileEndsWith) {
        this.fileEndsWith = fileEndsWith != null ? fileEndsWith.toLowerCase() : fileEndsWith;
    }
}

class ListenerList<L> {
    private List<L> listenerList = new ArrayList<L>();

    public interface FireHandler<L> {
        void fireEvent(L listener);
    }

    public void add(L listener) {
        listenerList.add(listener);
    }

    public void fireEvent(FireHandler<L> fireHandler) {
        List<L> copy = new ArrayList<L>(listenerList);
        for (L l : copy) {
            fireHandler.fireEvent(l);
        }
    }

    public void remove(L listener) {
        listenerList.remove(listener);
    }

    public List<L> getListenerList() {
        return listenerList;
    }
}

Use it on activity onCreate (directory selection option is commented):

protected void onCreate(Bundle savedInstanceState) {
    super.onCreate(savedInstanceState);
    File mPath = new File(Environment.getExternalStorageDirectory() + "//DIR//");
    fileDialog = new FileDialog(this, mPath, ".txt");
    fileDialog.addFileListener(new FileDialog.FileSelectedListener() {
        public void fileSelected(File file) {
            Log.d(getClass().getName(), "selected file " + file.toString());
        }
    });
    //fileDialog.addDirectoryListener(new FileDialog.DirectorySelectedListener() {
    //  public void directorySelected(File directory) {
    //      Log.d(getClass().getName(), "selected dir " + directory.toString());
    //  }
    //});
    //fileDialog.setSelectDirectoryOption(false);
    fileDialog.showDialog();
}