If you ask for an "easy" way to determine if a ligand is a pi acceptor/donor or none, look at the spectrochemical series. A quick-and-dirty rule (which means that it is correct most of the time but not always) is that weak ligands (iodide, bromide, hydroxide etc) are pi-donor ligands. The medium ligands (water, ammonia etc) are pi-neutral, and the strong ligands (cyanide, carbonyl, bipyridine, etc) are pi-acceptor.
But this method is some kind of reversing here. Because we normally use a ligand's pi character to guess if it is a strong or weak ligand.
If you can, ligand field theory is your best friend in this problem.
Generally, a pi-donor ligand has an extra electron pair that lies in an orbital that has approximately the same symmetry as the metal center's orbital (usually t2g). This allows the two orbitals to interact with each other and creating two new sets of orbitals: t2g and t2g* (as in the molecular orbital theory). The ligand's pi electron will reside in the lower t2g orbital, whereas the metal's original d-electrons will reside in the t2g* one. That explains your question of the copper complexes.
In the other hand, a pi-acceptor ligand has an empty molecular orbital of low energy and approximately the same symmetry as the orbital of the metal center. In this case, it will interact with the metal's orbitals and the d-electrons of the metal will be delocalized to the ligand. These ligands usually have a highly conjugated system that greatly decreases the energy of the LUMO (bipyridine, triphenylphosphine etc). In the cases of carbon monoxide (called carbonyl as a ligand) and cyanide, they have low-energy LUMO for other reasons.
A pi-neutral ligand (also called sigma-donor ligand) does not process any of the above characteristics. For example, ammonia has only one electron pair on the nitrogen, and that pair is already used in the formation of the sigma bond, so no more pi electrons to donate. The LUMO of the ammonia is also high in energy, so no pi accepting here.
I think trimethyl phosphate is mostly a pi-neutral with the donor atom is the oxygen double-bonded to phosphorus. The molecule is not conjugated at all, and the oxygen can only donate one pair of electron since the other's orbital does not have the same symmetry.