It's not just under ICS. There have ALWAYS been occasional issues when using chargers not really designed for the phone.
Really need to use the chargers that came with and/or designed to work with the Bionic.
With respect I would offer a counter point:
Any USB charger that matches the proper current specification should be sufficient really.
There's no voodoo in these USB chargers nowadays, just specs relating to how much current they deliver.
If you have an available charger that meets the spec, by all means use it!
Adhering strictly to a brand standard creates the potential for more needless e-waste IMO.
For instance, the Samsung charger I was using at work for nearly the last year can only output 700ma, as opposed to the Motorola charger which delivers 850ma. I used a 2.0 amp charger as well with success.
Ultimately, the issue I was having was with the charger I was using at work, but only in that it didn't meet the same current spec as the Motorola, not that it was a different brand.
It would seem that under GB a weaker charger would suffice, but under ICS a charger with a MINIMUM of 850ma output is required to support charging the battery while the Bionic is also engaging in more active processes.
Now that I'm aware I'll simply use a different charger that delivers at least 850ma.
I guess it stands to reason that the power usage of the phone is going to be altered by the OS having more access to the dual-core hardware.
Wimpy chargers just won't cut it anymore!
EDIT
As I plow through my collection of USB chargers it would seem that most of them are pretty standard, which is to say 500ma. So check those tiny labels!
I also haven't noticed any difference in different brands of micro USB cables, aside from the usual concerns with crappy quality ones.