  We establish a platform to transfer $L_p$-completely bounded maps on tensor
products of von Neumann algebras to $L_p$-completely bounded maps on the
corresponding amalgamated free products. As a consequence, we obtain a
H\"ormander-Mikhlin multiplier theory for free products of groups. Let
$\mathbb{F}_\infty$ be a free group on infinite generators $\{g_1,
g_2,\cdots\}$. Given $d\ge1$ and a bounded symbol $m$ on $\mathbb{Z}^d$
satisfying the classical H\"ormander-Mikhlin condition, the linear map
$M_m:\mathbb{C}[\mathbb{F}_\infty]\to \mathbb{C}[\mathbb{F}_\infty]$ defined by
$\lambda(g)\mapsto m(k_1,\cdots, k_d)\lambda(g)$ for $g=g_{i_1}^{k_1}\cdots
g_{i_n}^{k_n}\in\mathbb{F}_\infty$ in reduced form (with $k_l=0$ in
$m(k_1,\cdots, k_d)$ for $l>n$), extends to a complete bounded map on
$L_p(\widehat{\mathbb{F}}_\infty)$ for all $1<p<\infty$, where
$\widehat{\mathbb{F}}_\infty$ is the group von Neumann algebra of
$\mathbb{F}_\infty$. A similar result holds for any free product of discrete
groups.
