How quickly can we decompose a number into a set of residues?
If $m=2^k-t$ where $t$ is small, then you can reduce modulo $m$ faster.  In particular, $a \cdot 2^k + b \equiv at+b \pmod{m}$, so reducing an $\alpha k$-bit number modulo $m$ typically requires $\alpha$ multiplications by $t$ and $\alpha$ additions.  This can be significantly faster than the normal algorithms, if $t$ is small enough.

Something similar holds for $m=2^k+t$, too.

So, one plausible approach is to choose a modulus $m$ that is a product $m=m_1 m_2 \cdots m_n$ where each $m_i$ has the form $2^k \pm t$, where $k$ is the word size of your computer (e.g., $k=64$).