Deleting certain elements from a nested list

For the record, so that this question have answers in an Answer block, correct answers given in the comments to the OP include

• list/.{_, 0} :> Sequence[]
• DeleteCases[list, {_, 0}, ∞]

In addition, here are two that are similar to the ones above but slightly more specific to the requirements stated in the OP:

• list/.{_, 0} -> Sequence[] (Rule instead of RuleDelayed)
• DeleteCases[list, {_, 0}, {3}] (deletes only sublists at level 3)

Let's use a more complex expression than given above to stress the solution methods a bit more.

data = 
  {{{{a, 0}, {b, 1}, {c, 2}}, {{d, 4}, {e, 5}, {f, 6}}},
  {{{{{a, 0}}}, {b, 1}, {c, 2}}, {{d, 4}, {{e, 0}, {f, 6}}}}};

I think the following have advantages over the solutions already given.

data /. {_, 0} -> Nothing
DeleteCases[data, {_, 0}, {-2}]

Both give

{{{{b, 1}, {c, 2}}, {{d, 4}, {e, 5}, {f, 6}}}, 
 {{{{}}, {b, 1}, {c, 2}}, {{d, 4}, {{f, 6}}}}}

The first method I think more easily understood than /. {_, 0} :> Sequence[] and second simply is more robust -- DeleteCases[data, {_, 0}, {3}] fails on this more complex data list.