How dependent is Earth is with the other Planets' gravity? How would the Earth be affected if the Moon was destroyed as far as our orbit around the Sun? What if Mars or Jupiter was destroyed?
The orbital speed of the Moon around the Earth is 1022 m/s. The Earth is 81 times more massive, than the Moon. Thus, a sudden disappearance of the Moon would cause an $\frac{1022}{81}=12 \frac{m}{s}$ change of the Earth's orbit, which would be negligible.

The disappearance of the Mars or Jupiter would have yet lesser effect.