I think it's easier to understand, if you understand the underlying issue first.
Japanese games don't checked for region detection bit. There is a bit on the system for JP and Other, and games can read this bit to identify the region of the system. 'Other' happens to be North America (Europe test run of the TG doesn't counter, but I believe it's also set to Other).
The only games that I know of, that check this region bit, are US games. And those are US hucards, not CD games. There's also another method of region protection between the JP and Other region models; the US/Eu systems have the data bus pins in a different physical order. Since Japanese hucard games DON'T check for the region bit, they could normally run on a US/EU system - except for that one small thing that they changes, those 8 data lines on the hucard port. So it keeps US/NA hucards from playing on japaense systems, and vice versa. Even with a flash card, you either have to mod your Japanese system for the region bit or patch every games to bypass it; which ever (for using US hucards on Japanese systems).
CD games don't check for this. Either US/NA or Japanese ones. Both can be played on either system. I think there are 2-3 games total, CD, that don't run on US models simply because of a bug. One of them is Altered Beast, and that's simply a bug in the software when accessing a system card other than the original 1.0 (even on Japanese systems).
Where you get in trouble with CD games and different region CD units, is the system cards themselves. Obviously you can just run a JP system card on a US model, and vice versa, as-is (because system cards are hucards).
If you're buying or have an 'all-in-one unit', then it's usually only the Arcade Card that's an issue; assuming you have a US/NA model. You'll have to mod the system with a bus changing switch, or buy a converter pass-through card. I've heard of problems with pass-through when using it on the Arcade Card, but nothing conclusive. If you have an all-in-one unit (or a SuperCDROM^2 addon), get the Duo version of the Arcade Card. There are less parts to 'fail' inside it, and probably draws a little less juice from the hucard port (which is a problem for some systems and/or weakened AC/DC power units over the years).
As thesteve said, if you have something like the Turbo Everdrive, then you can run older system cards on all-in-one/supercdrom^2 units. People do this to play Altered Beast, but also to access the system card error screens (sometimes funny animation sequence or mini games like in Rondo of Blood).