Dr. Entropy is right, Jim. I would suspect the gulp valve. From what I've read they are of a very poor design and prone to failure.
Don't know what state you're in and, consequently, have no way of knowing if you can legally remove the emissions stuff. But if you can, do yourself a big favor and do so. I removed my emissions stuff (air pump, injectors, and gulp valve) about 3 years ago and the backfiring I had been experiencing all went completely away.
If you'll look at many MGB engines that show up periodically on these forums, you will note that the emissions stuff on most has been removed. Let's face it - MG only came up with that stuff (emissions control system)
in order to comply with U.S. regulations at the time. I doubt the MG people really put a lot of engineering know-how behind this effort.