In terms of generating a depolarizing or hyperpolarizing graded potential on a postsynaptic cell, what do you think will be the most important factor?

The type of ion channel opened will be the most important factor.

Explanation:

At certain occasions a given neurotransmitter can combine with a K+ chemically gated channel and at certain occasions the same neurotransmitter can combine with a Na+ chemically gated channel. Thus, the same neurotransmitter can produce depolarization if it opens the Na+ channel and a hyper-polarization if it opens the K+ channel.  

In these situations, the structural make-up and sensitivity of the postsynaptic cell really determines if the neurotransmitter will inhibit (depolarize) or excite (depolarize) the postsynaptic cell. It is more essential to know the kind of channel, which opens than the particular identity of the neurotransmitter.