If you know, in advance, that the state you want to deamplify is specifically $|000\rangle$, there are a couple of strategies that you could follow.

For example, introduce an ancilla and perform the multi-controlled not, targeting the ancilla, where it is controlled off every qubit in the original state being in the $|0\rangle$ state. So, you'd be doing
$$
\frac{1}{\sqrt{8}}\sum_x|x\rangle|0\rangle\mapsto \frac{1}{\sqrt{8}}\left(|000\rangle|1\rangle+\sum_{x\neq 000}|x\rangle|0\rangle\right).
$$

If you want to completely get rid of the $|000\rangle$ term, just measure your ancilla in the standard (0/1) basis. If you get the 0 answer, you've succeeded (here, this happens with probability 7/8). If you get the 1 answer, you've failed. You produce your state again and repeat until success.

If all you want to do is decrease the amplitude of $|000\rangle$ rather than completely remove it, you can correspondingly increase your probability of success. A simple strategy is to apply the POVM/filtering operation
$$
\left(\begin{array}{cc} 1 & 0 \\ 0 & \alpha \end{array}\right)
$$
for $\alpha<1$. Afterwards, you'd repeat your multi-controlled-not operation to disentangle the ancilla.

These strategies are perhaps conceptually simpler to understand than amplitude amplification, and work well if the amplitude you're trying to de-amplify is small enough. However, if the amplitude you're trying to amplify is too small, you won't do any better than the scaling resulting from amplitude amplification, and you'd be much better off seeing if you can change the circuit that's producing your state so that you don't have so much of the $|000\rangle$ component in the first place. For instance, in your example, instead of applying $H^{\otimes 3}$, you'd find a slightly more complicated circuit such as
where $U|0\rangle=(\sqrt{3}|0\rangle+2|1\rangle)/\sqrt{7}$ and $V|0\rangle=(|0\rangle+\sqrt{2}|1\rangle)/\sqrt{3}$.

2I'm not clear why you don't want to use amplitude amplification, which does exactly what you are asking for – glS – 2019-08-01T11:22:41.697

@glS The reason I don't want to use amplitude amplification is that I want to de-amplify one specific, constant state and I want to avoid the $O\left(\sqrt{N}\right)$ time required to do that completely. – Woody1193 – 2019-08-02T04:52:10.530