Is there any evolutionary advantage to finding melodies or harmonies pleasurable? Does the ear pick up these particular oscillating waves differently from other sounds, and if so, how does that affect our perception of pleasure? I'm looking for some sort of signalling pathway (most likely involving neurotransmitters I realize).
There are strong connections between the auditory cortex and the limbic system, which includes such structures as the hippocampus and the amygdala.
A recent paper  builds on earlier notions of emotional "significance" of music without any lyrics. It adds in lyrics, so giving a perspective of which portions of the brain are reacting to which component of the music.
Additionally, contrasts between sad music with versus without lyrics recruited the parahippocampal gyrus, the amygdala, the claustrum, the putamen, the precentral gyrus, the medial and inferior frontal gyri (including Broca’s area), and the auditory cortex, while the reverse contrast produced no activations. Happy music without lyrics activated structures of the limbic system and the right pars opercularis of the inferior frontal gyrus, whereas auditory regions alone responded to happy music with lyrics.
One of the limitations of this particular study is that the subjects self-selected their own pieces, which may limit the reliability of the results. Of course, defining "happy" or "sad" for every individual is slightly subjective and difficult. They cited an earlier "pioneering" study which standardized the musical selection between subjects. Without consideration of the lyrics:
The first pioneer study using functional magnetic resonance imaging (fMRI) by Khalfa et al. (2005) chose a controlled manipulation of two musical features (tempo and mode) to vary the happy or sad emotional connotations of 34 instrumental pieces of classical music, lasting 10s each. Sad pieces in minor-mode contrasted with happy pieces in major mode produced activations in the left medial frontal gyrus (BA 10) and the adjacent superior frontal gyrus (BA 9). These regions have been associated with emotional experiences, introspection, and self-referential evaluation (Jacobsen et al., 2006; Kornysheva et al., 2010).
As an aside to answer your final thought, in cases like this I think trying to jam everything under an umbrella of one "neurotransmitter system" or another can make things overly simplistic to the point where you lose focus of the diversity of receptors expressed. You can say a system is driven by Dopamine, but D1 and D2 receptors have exactly the opposite effects on the neuron.
 Brattico, E., Alluri, V., et al (2011) A functional MRI study of happy and sad emotions in music with and without lyrics. Frontiers in Psychology, 2: 308. doi: 10.3389/fpsyg.2011.00308 (free pdf)
(see also, http://www.sciencedirect.com/science/article/pii/S0028393206003083 and related)
In music, harmonies are simultaneous combination of tones or chords that are concordant.
In physics, each note is actually a vibration with defined wavelenght, the concordance can be explained in mathemathical terms, for instance with regard to coincidence of phase oscillation.
In physiology, the ear perceives air vibrations and send them to the brain by means of trains of pulses.
According to some scientists, a music providing regular train of pulses (like harmonic music and rhytmic music) should be more pleasing, probably because of stimulation of the limbic system, as the other answer explains.
Source: Ushakow et al. 2011, Physical Review Letters, DOI 10.1103/PhysRevLett.107.108103
Lay explanation: Why harmony pleases the brain, New Scientist, Sept 2011