Do only black holes emit gravitational waves?
Gravitational waves (GW) are emitted by all systems which have an 'accelerating quadrupole moment' --- which means that the systems have to be undergoing some sort of acceleration (i.e. a constant velocity is not enough), and they have to be asymmetric. The perfect example is a binary system, but something like an asymmetric supernovae is also expected to emit GW.
The total mass of the system doesn't matter [1] in determining whether GW are produced or not. It does determine how strong the GW are. The more massive the system and the more compact they are, the stronger the GW, and the more likely they are to be detectable---of course, how often an event happens nearby is also very important. The examples you give, black holes (BH) and neutron stars (NS), are some of the best sources because they are the most compact objects in the universe.
Another aspect to consider is the detection method. LIGO for example is only sensitive to GW in a certain frequency range (kilohertz-ish), and roughly stellar-mass systems (like binaries of NS and stellar-mass BH) emit at those frequencies. Something like supermassive BH binaries, in wide-separation orbits, emit GW at frequencies of (often) nanohertz --- which are expected to be detected by an entirely different type of method: by Pulsar Timing Arrays.
There is a proposed mission called the Laser-Interferometer Space Antenna (LISA) which would detect objects at frequencies intermediate between Pulsar Timing Arrays and ground-based interferometers (like LIGO), which would detect tremendous numbers of White-Dwarf binaries.
[1] General Relativity (GR), the theory which describes gravity and gravitational waves, has a property called "scale invariance". This means that no matter how massive things are, all of the properties of the system look the same if you scale by the mass. For example, if I run a GR simulation of a 10 solar-mass BH, the results would be identical to that of a 10 million solar-mass BH --- except one million times smaller in length-scales (for example the radius of the event horizon). This means that no matter the total mass of the binary, GW are still produced. It's also very convenient for running simulations... one simulation can apply to many situations!
Gravitational waves are emitted by all masses with accelerating gravitational quadrupole moments, but rarely with sufficient power to be detectable. I will restrict my answer to dealing with merging binary systems, but similar considerations apply to other scenarios just on dimensional arguments.
The power emitted by gravitational waves from a pair of orbiting masses is given by $$ P = 1.7\times 10^{54} \frac{M_1^2M_2^2(M_1+M_2)}{R^5}\ \ {\rm W},$$ where $M_1, M_2$ are the masses of the two components in solar masses and $R$ is the separation of the two masses in kilometeres. The frequency of the gravitational waves produced occurs at twice the orbital frequency.
To put that in perspective, the largest of the two recent LIGO detections turned 3 solar masses into gravitational wave energy in $\sim 0.2$ s, emitting an average power of $\sim 3 \times 10^{48}$ W. This arose from a pair of 30 solar mass black holes, separated by a few times their Schwarzschild radii (say $4 \times 2 GM/c^2 = $ 360 km). Plugging these numbers into the formula above suggests $P \sim 10^{49}$ W, similar to the estimate based on the mass discrepancy between the black holes before and after they merged. This event was just detectable by LIGO.
All orbiting pairs of masses give off gravitational waves in this manner. But their masses and orbital separations do not result in significant (detectable) energy losses through gravitational wave emission due to the steep dependencies on mass and separation.
Black holes were once massive stars. In fact they were even more massive stars, since a black hole progenitor loses mass during its lifetime. The reason that black hole binary systems are favourable for gravitational wave detection is that they can get close together before they merge. i.e. There are plenty of stars out there with huge component masses but they cannot be brought close enough together to produce detectable gravitational waves without them first merging. The radius of a typical "normal" star is 5 orders of magnitude larger than the Schwarzschild radius for a black hole of similar mass. Looking at the formula this means the gravitational waves produced by such a system would be 25 orders of magnitude smaller than ifsimilar mass black holes were merging.
Neutron stars represent an intermediate case. Whilst their radii and hence closest possible orbits are only $\sim 3$ times larger than for black holes, their masses are limited to around $\leq 2 M_{\odot}$. So compared with the 30 solar mass black holes mentioned above this means that the emitted power from a pair of merging $1.5 M_{\odot}$ neutron stars would be down by $\sim 2$ orders of magnitude and so they would only be detectable if they were closer to the Earth by factors of 10.
The crucial parameter here is $(M/R)^5$. If we work in a natural set of units then because the Schwarzschild radius is proportional to mass we can say that if all black holes have $M/R \simeq 1$ (forgetting about spin for a moment), then for a neutron star, $M/R \sim 0.4$ and the power emitted is only $(M/R)^5 \sim 0.01$ that of a pair of equivalent mass black holes. For a normal star like the Sun $M/R \sim 4\times 10^{-6}$ and so the power falls by a factor $\sim 10^{-27}$.
Interestingly, considerations like these suggest that binary black holes of any mass should produce roughly the same power in gravitational waves as they reach the point of merger. However, the waves are produced at very different mass-dependent frequencies. A rule of thumb is that the peak frequency will occur at $\sim \sqrt{G\rho}$, where $\rho \sim 3(M_1+M_2)/4\pi R^3$ is the average density. For a pair of 30 solar mass black holes separated by their Schwarzchild radii $\sqrt{G\rho} \simeq 500$ Hz.
This is bang in the centre of the most sensitive part of the frequency spectrum for the LIGO detectors. Less massive black hole binaries will produce higher frequencies ($\propto M^{-1}$); supermassive black hole mergers or binaries with components with lower $M/R$ will produce gravitational waves at way below the frequencies to which LIGO is sensitive, but for which space-borne interferometers are currently being designed.