It depends on the curse
You must look at an actual item to find out what happens. And how effective leaving it on the ground would be.
Lets look at the Arrow-Catching Shield:
This shield is cursed. Attuning to it curses you until you are
targeted by the remove curse spell or similar magic. Removing the
shield fails to end the curse on you. Whenever a ranged weapon attack
is made against a target within 10 feet of you, the curse causes you
to become the target instead.
It very specifically states removing the shield has no effect on the curse. So you could drop it on the ground, but your bear would still have missiles chasing it.
And the Stone of Ill Luck (Tales from the Yawning Portal, p. 229):
Curse. This item is cursed. While it is on your person, you take
a −2 penalty to ability checks and saving throws. Until the curse is
discovered, the DM secretly applies this penalty, assuming you are
adding the item’s bonus. You are unwilling to part with the stone
until the curse is broken with remove curse or similar magic.
You would simply be unable to choose to drop the item when wildshaping, as you are unwilling to do so.
Basically, it takes more than physically removing an item to end a curse. Some literally are unable to be removed (being more specific than wildshape, it would take precedence), some you are unwilling to remove, and others merely continue with or without the item.
Ending the attunement
Fortunately, there is a big benefit in leaving a cursed item on the ground. You cannot willingly end the attunement; however, if you look at the attunement rules, there are ways to unwillingly end attunement:
A creature’s attunement to an item ends if the creature no longer
satisfies the prerequisites for attunement, if the item has been more
than 100 feet away for at least 24 hours, if the creature dies, or if
another creature attunes to the item.
So you could be “forced” to break attunement if you move away from the item, or if some other hapless victim finds it and attunes to it. Some DMs may consider the act of leaving it over 100 feet away as voluntary and therefore not allow that to break the attunement.
Keep in mind that even if you break the attunement, it doesn’t automatically end the curse. It does however at least allow you to attune to a new (hopefully not cursed) item in the meantime.