How can one travel with X-Ray sensitive equipment or electronic devices?

The FAA regulation has this to say:

If requested by passengers, their photographic equipment and film packages shall be inspected without exposure to an X-ray system.

I am unaware of any official TSA word on this but it'd be very hard to imagine them contradicting the FAA.

Anything is a "photographic equipment" if you try hard enough. First ask politely not to be x-ray'd and if they don't then pull out the printout of the regulation and ask them to kindly please attend to federal regulations. Avoid trying to explain what it is. Only if pressed further tell them it's photographic equipment but don't go into details much. Don't forget the TSA is essentially a federal jobs program for otherwise unemployable people. The chances of them understanding what is an FGA and why you don't want to X-Ray it is nil. It's photographic equipment, like an external timer. Use simple words of one syllable or less.

Edit: to react to @reirab's comment, finding TSA regulations is very difficult. However, 49 CFR 1546.209 - Use of X-ray systems. seems to apply to the TSA (and foreign air carriers but that's not relevant):

(2) At locations at which a foreign air carrier or TSA uses an X-ray system to inspect checked baggage the foreign air carrier must ensure that a sign is posted in a conspicuous place where the foreign air carrier accepts checked baggage.

(3) The signs required under this paragraph must notify individuals that such items are being inspected by an X-ray and advise them to remove all X-ray, scientific, and high-speed film from accessible property and checked baggage before inspection. This sign must also advise individuals that they may request that an inspection be made of their photographic equipment and film packages without exposure to an X-ray system.

Please note how this copies the FAA regulation word-by-word starting from "their". And then this repeats:

(4) If requested by individuals, their photographic equipment and film packages must be inspected without exposure to an X-ray system.

OK, So I've never tried this with electronic equipment (which I think will make it harder).

But I've flown several times with personal radiation dosimeters as required by my employment at the time. Obviously, an x-ray exposure would contribute to a recorded dose - overreporting is a much smaller problem than underreporting, but still something to be avoided.

I've been able at all airports that I've tried, to either leave the item in my pocket, or where clearing of pockets was required (most often where terahertz imaging was used), hand it to a member of security staff to bypass the x-ray machines. Recently, this request to bypass x-ray has always led to a chemical swab being taken of the dosimeter (presumably policy, because it's too small to be a dangerous device).

This has sometimes involved a minute or two of explanation to the security staff in question. I've done this at airports in the US, UK and France. The UK staff at Manchester were perhaps the hardest to convince, and remained somewhat suspicious. TSA staff at IAD took some time to explain, but were friendly once they understood (they were also sticklers for "nothing in pockets", not even allowing card voucher type receipts etc).

So in principle, explaining that something you are carrying cannot be exposed to x-rays can get it to avoid the x-ray machines. However, it possibly helps if the item is small and safety-relevant, and I'm not sure I'd chance it with something that I couldn't let be xrayed. (Of course, the dosimeter also records a small level of cosmic ray exposure from long haul flights. So you should think about that with anything you are flying with).

For the United States, items are screened by the Transportation Security Administration. While I didn't find anything on their website stating a general policy for all x-ray-sensitive equipment, they do say this regarding camera film (bold emphasis in original, italics are mine):

Undeveloped camera film is not prohibited, but you should only transport it in your carry-on baggage; the equipment used to screen checked baggage may damage undeveloped film.

If you are transporting high speed (800 ISO and higher) or specialty film, you may request to have it physically inspected when presented at the screening checkpoint instead of undergoing x-ray screening. You may also request that all of your undeveloped film be physically inspected instead of undergoing x-ray, particularly if your film has or may be screened by x-ray more than five times. To facilitate physical inspection, remove your undeveloped film from the canister and pack it in a clear plastic bag. We recommend leaving your film in the unopened manufacturer’s packaging.

Even if an item is generally permitted, it may be subject to additional screening or not allowed through the checkpoint if it triggers an alarm during the screening process, appears to have been tampered with, or poses other security concerns. The final decision rests with TSA on whether to allow any items on the plane.

Source: The "When I fly can I bring my..." search result for "camera film" on TSA's website.

They may or may not treat your equipment in the same way as camera film, but this was the closest I was able to find on their website.

I would especially advise paying attention to the last line in their statement. Effectively, TSA has no obligation to let you bring any given item through the checkpoint. A particularly non-tech-savvy and/or particularly suspicious agent or simply an agent who is in a bad mood might refuse to allow a given item to pass through the checkpoint without going through the x-ray scan.

Most agents will probably be more reasonable than that and will generally work with you as long as you're polite and cooperative, but there's no guarantee that all of them will. If you want to be really sure a given item is not subjected to an x-ray scan, it's probably better to transport it another way.

TSA says they will do hand-checks on any equipment that is sensitive to X-rays. I have never put it to the test, but you might consider mailing the thing rather than take the chance.