I have been using Eneloop rechargeable batteries in my remotes for almost a year now. It's pretty awesome. Unlike NiMH or Li-Ion, they still have ~80%+ of their life after being charged and left on a shelf for 6 months. So they are a perfect application for this type of thing.
Only downside is that they're 1.2v and don't have enough power for some applications (like my lighted & battery-powered pepper mill. It was a gift.).
The shake&wake batteries are still awesome! I would love to have them. But Eneloop is here now, and very cost-effective.
Non-rechargeable 1.5v batteries only start at 1.5v. As you use them up, the voltage drops, going down to about 1v just before the battery poops out. The curve is not linear.
These batteries are down to 1.2v around 30-50% of the way through their life.
The rechargeable batteries start around 1.2v, but their discharge curve is much more level. They stay near 1.2v right up to the end.
If a device has trouble with rechargeable batteries because of the voltage, that device is also going to have problems with non-rechargeables, as the non-rechargeables spend at least half their service life below 1.2v.
The rechargeable batteries can confuse battery meters on some devices, though. If the device assumes non-rechargeable batteries, and then tries to estimate remaining battery by comparing the voltage to a typical non-rechanrgeable discharge curve, and you put rechargeable batteries in, the device is going to think they are starting out at 50-80%, and then is going to think they are draining very very slowly.