Some iPad 2 Smart Covers from third party manufacturers will not work with the new iPad. The third generation iPad has a slightly different magnet configuration that may prevent the iPad from sleeping/waking with the third party case.
Even older official Apple Smart Covers might not work with the iPad 3. Those manufactured in October 2011 or later should work fine. Apple has switched the polarity of the sleep/wake switch magnet, to prevent this from being activated accidentally by another magnet.
If your case worked fine with the iPad 2 but now it won't put the new iPad to sleep, this is likely the problem. Some case manufacturers, such as Targus, have offered to replace non-working Smart cases free of charge for iPad 3 owners.
The new iPad is also slightly thicker than the iPad 2, so some snug iPad 2 cases might simply not fit on the new tablet.