(Turning my comments into an answer, as per Yemon's suggestion).
The case where $\mu$ is purely atomic is essentially the only case where a Pitt-type theorem holds for operators $L_p(\mu)\longrightarrow L_q(\mu)$, where $1\leq q,p<\infty$. Indeed, let us consider, say, the case where $\mu$ is Lebesgue measure on $[0,1]$ (all other non-purely atomic cases should reduce to this one). By the assertions of Yemon's first comment, one wants to show that there is a noncompact operator $L_p(\mu)\longrightarrow L_q(\mu)$. To this end I suggest use the fact that for $1\leq r<\infty$ the closed linear hull $R_r$ in $L_r(\mu)$ of the sequence of Rademacher functions on $[0,1]$ is isomorphic to $\ell_2$, and $R_r$ is complemented in $L_r(\mu)$ if and only if $1<r<\infty$ (a good reference for this result is Albiac and Kalton's book Topics in Banach space theory; there it is stated as Proposition 6.4.2). The time to treat the cases $p=1$ and $p>1$ separately has now arrived.
For $p=1$, we let $X$ be a complemented subspace of $L_1(\mu)$ isomorphic to $\ell_1$ and let $P$ be a projection of $L_1(\mu)$ onto $X$. Since every separable Banach space is the range of a continuous linear operator from $\ell_1$, there exists a surjective operator $T:X \longrightarrow L_q(\mu)$. The map $x\mapsto T(Px)$ is a surjective (hence noncompact) linear operator of $L_1(\mu)$ onto $L_q(\mu)$.
For $p>1$, let $Q$ be a projection of $L_p(\mu)$ onto $R_p$ and let $U: R_p\longrightarrow$ be an isomorphic embedding onto $R_q$. The map $x\mapsto U(Qx)$ is a linear operator of $L_p(\mu)$ onto the infinite dimensional closed subspace $R_q$ of $L_q(\mu)$; this operator is, in particular, noncompact.