Join / merge lines that touch and intersect in QGIS

In QGIS Plugins you'll find a 'merge lines' plugin, which at first sight seems to accomplish what you are after.

cited from description:

MergeLines

Simplifies the topology of a line network by merging adjacent lines

This plugin merges segments of a line network (e.g. river network) in order to simplify its topology. Two merging methods are currently available : length (a segment is merged with its longest neighbor) and alignment (a segment is merged with its best aligned neighbor).

UPDATE:

Below you find a geoprocessing script of which I hope that it does what you want. For testing purposes I created a shapefile with a bunch of irregularly intersecting lines and no attributes (network):

The standard dialog when executing the script looks like this (in this case the result is a memory layer):

Running the script produces a 'copy' of the input data with a field 'subnet' distinguishing to which subnet a feature belongs. With a categorized style the result looks like this:

This can be dissolved using the field 'subnet'.

Create a new geoprocessing script, copy the code in the editor, save it and things should work.

##Networking=group
##lIn=vector
##lOut=output vector

from qgis.core import *
from qgis.utils import *
from qgis.gui import *
from PyQt4.QtCore import *
from PyQt4.QtGui import QColor
#from processing.core.VectorWriter import VectorWriter

c=iface.mapCanvas()
rubberBand = []

v = processing.getObject(lIn)

#generate a list of all features in the layer
subNets = []
flist = []
for f in v.getFeatures():
    flist.append(f)


def makeSubnets(featureList):
    #print "making subnet ---------------------------------------------------"
    if len(featureList) > 1:
        print [featureList[0]]
        print featureList[1:]
        #print "start finding a subnet"
        fTest, fRest = findSubnet([featureList[0]], featureList[1:])
        #print "finished finding a subnet"
        subNets.append(fTest)
        makeSubnets(fRest)
    else:
        subNets.append(featureList)

def findSubnet(featTest, featRest):
    found = True
    while found:
        newTestList = []
        #print "candidates: ", len(featTest)
        #print "search in: ", len(featRest)
        #print "-------------"
        for fT in featTest:
            for fR in featRest:
                if not fT.geometry().disjoint(fR.geometry()):
                    #print "!"
                    newTestList.append(fR)
                    #addRubberBand(fR.geometry())

        featTest += newTestList

        if newTestList == []:
            found = False
        else:
            #print "Found (undis)joining segments"
            for fn in newTestList:
                if fn in featRest:
                    featRest.remove(fn)
                    #print "removed ", fn
                else:
                    pass
                    #print "allready removed ", fn

    return featTest, featRest

def addRubberBand(theGeom):
    rubberBand.append(QgsRubberBand(c, False))
    rubberBand[-1].setToGeometry(theGeom, None)
    rubberBand[-1].setColor(QColor(255,0,0))
    rubberBand[-1].setWidth(3)

makeSubnets(flist)

fields = QgsFields()
fields.append(QgsField('subnet', QVariant.Int))
writer = QgsVectorFileWriter(lOut, None, fields, QGis.WKBLineString,     v.crs())

net = 0
for sn in subNets:
    for f in sn:
        #print net, f
        feat = QgsFeature()
        feat.setFields(fields)
        feat.setGeometry(f.geometry())
        feat.setAttribute('subnet', net)
        writer.addFeature(feat)
    net += 1
del writer

UPDATE #2:

To create a geoprocessing script do the following (I've got the german gui, so I try to translate in en):

A: Menu 'Processing' -> 'Toolbox' (appears as a dock on the right)

B: Under 'Scripts [...]' -> 'Tools' doubleclick 'create new script'

An Editor with a little toolbar appears, in wich you copy the code above. Herein you can:

C: Save the script. It appears (in this case) in the group 'Networking' or in whatever group you write in the first line of the script ##MyGroup=group. Be aware not to write blanks in the ##-lines!!!

D: Start the script with the two little gears. A gui appears (cp. above) with the in- and output layers defined in the script line 2 and 3. When saved, alteratively start the script by doubleclicking its name under 'scripts' > 'mygroup' > 'myscriptname' (if saved under myscriptname.py)