What do you mean by seven canonical hours? Do you mean the four hours-type-prayers and then vespers, etc.?
Yes. Vespers, Compline, Midnight, Matins, and the 3rd, 6th, and 9th Hours. Not every tradition has a 1st Hour, but if you do, that too.
That said, it's not simply or merely a matter of reaching an hour, whipping out five books, and rattling off 75 min worth of prayers. There is something to be said for these hours as "moments" of prayer, which is where the abbreviations come in (see below).
I've also never heard anyone say that every Christian was obliged to--in fact, I've had priests tell me not to say them. Not that I'm saying you pulled this out of thin air, just saying that I've been taught differently.
I can't say why priests instructed you not to pray the hours. My guess is that it may be a sort of default advice they give to people in order to keep them from taking on more than they can handle and then burning out. Another reason for thinking this is the fact that the Byzantine services require so many books, a knowledge of how to put things together accurately, etc., that it would be next to impossible for most people to do Vespers and Matins on their own, though the other services aren't nearly as challenging.
Regarding the obligation, it is indicated in the canons. I would quote them here, but I don't have them in English, so I'd have to find the book in the disorganised pile of books I call my library, locate them, translate them, and post them. If necessary, I will, but take my word for it for now.
Is this obligation talked about, or even done, in Oriental Orthodox circles? It seems like the hours wouldn't be overly difficult... maybe 40 minutes for them... but all the other stuff? Well hey, if you can muster the will power to pray 3+ hours a day, good on you
In another thread (in the OO section), I spoke a bit about this from the perspective of our tradition. The canons I have in mind variously speak of the obligation to pray twice a day (evening and morning), thrice a day (evening, morning, and noon), and seven times a day. The traditional way of harmonising those canons is to allow the aggregation of the seven services into blocks of two or three. Furthermore, there are approved methods of abbreviating these hours to make them more manageable for those without as much time. The basic "lay" prayer books contain the hours abbreviated in two major variations. So yes, anyone who uses one of those books for their daily prayers is "saying the hours", even if they don't necessarily think of it in those terms. But those who can use the "official" books do use them and say the full office daily: it doesn't take 3+ hours a day. Supposedly, I pray too slowly, but when I do them, the Midnight Office takes about 45 min, and the morning and evening aggregates (Matins, 3rd, 6th; 9th, Vespers, Compline) each take about 30 min. That's still a lot for some, but the Church blesses a few "shortcuts", and I'll use them when/if necessary. And again, you always have the liberty of talking to your confessor and getting further instructions. The absolute minimum, barebones, nothing-less-than-this-counts abbreviation is simply the Trisagion prayers and the Creed: 4-5 minutes. Easy peasy.