Is there a way to show road directions in Google Map API v2?

Try this solution here. You can get driving or walking direction on V2.

The Overlay is indeed something to forget.

Polylines can easily be drawn

Just loop through yourr points after you parsed tjhe JSON response:

PolylineOptions rectOptions = new PolylineOptions()
        .add(new LatLng(37.35, -122.0))
        .add(new LatLng(37.45, -122.0))  // North of the previous point, but at the same longitude
        .add(new LatLng(37.45, -122.2))  // Same latitude, and 30km to the west
        .add(new LatLng(37.35, -122.2))  // Same longitude, and 16km to the south
        .add(new LatLng(37.35, -122.0)); // Closes the polyline.

// Set the rectangle's color to red

// Get back the mutable Polyline
Polyline polyline = myMap.addPolyline(rectOptions);

Your question title is much more general than your requirements, so I will answer this in a way that I think will benefit those viewing this question and hopefully meet your requirements in perhaps a different way.

If you are not showing directions in the context of a map already being a loaded fragment and something having been done to show directions over the map (which is probably similar to what the OP is doing), it's easier and I believe standard to do this with an Intent.

This launches a map pathing activity (through a separate application - where the app launched depends on the user's compatible apps, which by default is Google Maps) that plots directions from the origin address (String originAddress) to the destination address (String destinationAddress) via roadways:

// Build the URI query string.
String uriPath = "";
// Format parameters according to documentation at:
String uriParams = 
    "?api=1" +
    "&origin=" + originAddress.replace(" ", "+")
        .replace(",", "") +
    "&destination=" + destinationAddress.replace(" ", "+")
        .replace(",", "") +
Uri queryURI = Uri.parse(uriPath + uriParams);
// Open the map.
Intent intent = new Intent(Intent.ACTION_VIEW, queryURI);
startActivity(activity, intent, null);

(Where activity is simply the currently active Activity - obtained through whatever means are appropriate in the current programming context).

The following code gets an address String from a LatLng object (which must then be processed for the URI query String as above):

* Retrieves an address `String` from a `LatLng` object.
private void getAddressFromLocation(
    final StringBuilder address, final LatLng latlng) {
    // Create the URI query String.
    String uriPath = 
    String uriParams = 
        "?latlng=" + String.format("%f,%f",
            latlng.latitude, latlng.longitude) +
        "&key=" + GOOGLE_MAPS_WEB_API_KEY;
    String uriString = uriPath + uriParams;
    // Issue the query using the Volley library for networking.
    RequestFuture<JSONObject> future = RequestFuture.newFuture();
    JSONObject response = null;
    // Required for JsonObjectRequest, but not important here.
    Map<String, String> jsonParams = new HashMap<String, String>();
    JsonObjectRequest request = 
        new JsonObjectRequest(Request.Method.POST, 
            new JSONObject(jsonParams),
            new Response.Listener<JSONObject>() {
                public void onResponse(JSONObject response) {
                    try {
                        if (response != null) {
                            String resultString =
                            // Assumes `address` was empty.
                        } // end of if
                        // No response was received.
                    } catch (JSONException e) {
                        // Most likely, an assumption about the JSON 
                        // structure was invalid.
                } // end of `onResponse()`
            }, // end of `new Response.Listener<JSONObject>()`
            new Response.ErrorListener() {
                public void onErrorResponse(VolleyError error) {
                    Log.e(LOG_TAG, "Error occurred ", error);
    // Add the request to the request queue.
    // `VolleyRequestQueue` is a singleton containing
    // an instance of a Volley `RequestQueue`.

This request is asynchronous, but it can be made synchronous. You will need to call toString() on the actual parameter passed to address to obtain originAddress.