We establish sufficient conditions for the Marcinkiewicz-Zygmund type weak law of large numbers for a linear process $\{X_k:k\in\mathbb Z\}$ defined by $X_k=\sum_{j=0}^\inftyψ_j\varepsilon_{k-j}$ for $k\in\mathbb Z$, where $\{ψ_j:j\in\mathbb Z\}\subset\mathbb R$ and $\{\varepsilon_k:k\in\mathbb Z\}$ are independent and identically distributed random variables such that $x^p\Pr\{|\varepsilon_0|>x\}\to0$ as $x\to\infty$ with $1<p<2$ and $\operatorname E\varepsilon_0=0$. We use an abstract norming sequence that does not grow faster than $n^{1/p}$ if $\sum|ψ_j|<\infty$. If $\sum|ψ_j|=\infty$, the abstract norming sequence might grow faster than $n^{1/p}$ as we illustrate with an example. Also, we investigate the rate of convergence in the Marcinkiewicz-Zygmund type weak law of large numbers for the linear process.