Hi Stereo-guy, and a warm welcome to Audioholics.
I think you meant the rear (center back) surrounds, not the back effects channels.
* Anyway, if I was you, I'll call around to see if you can find an audio shop that will look into your Integra DTR-9.3 Receiver, for a free estimate. There are some, you just have to find out which one.
Do you have any friends that are handy with electronics?
Your receiver is a very nice one, it's worth fixing in my opinion, as I'm sure that the channel in question (the right one), should be easily fixed.
Too bad you don't live near by.

Even if it would cost you $100 or so, you'll still save a bunch, instead of buying an external amplifier.
Could be as simply replacing an output transistor, or even a burned fuse.
A fuse is about .99 cents, or maybe less. Could even be a bad solder joint right at that speaker terminal. Or a bad connection from a wire.
I know that you can do with an external amp, as your DTR-9.3 has the requisite preouts. And I can find you a powerful stereo amp for less than $200.
But it is my sincere belief that you can get your DTR-9.3 fixed for less than that, if you find the right guy that knows about electronics, living near by.
Call around, and each time you talk to a new audio repairman, ask him if he knows someone else that do free estimates.
Also, talk to several of your friends, I'm sure you'll find someone that knows someone of their friends with the knowledge on fixing receivers like yours with that channel interrupted.
Best of luck & please stay in touch to let us know about what was the exact problem and the fix for it.
Bob