Let $G_{(m,3,r)}=\langle x,y\mid x^{m}=1, y^{3}=1,yx=x^{r}y\rangle $ be a metacyclic group of order $3m$ , where ${\mathrm{ gcd}}(m,r)=1$ , $1 and $r^{3}\equiv 1$ (mod $m$ ). Then, left ideals of the group algebra $\mathbb {F}_{q}[G_{(m,3,r)}]$ are called left metacyclic codes over $\mathbb {F}_{q}$ of length $3m$ , and abbreviated as left $G_{(m,3,r)}$ -codes. A system theory for left $G_{(m,3,r)}$ -codes is developed for the case of ${\mathrm{ gcd}}(m,q)=1$ and $r\equiv q^\epsilon $ (mod $m$ ) for some positive integer $\epsilon $ , only using finite field theory and basic theory of cyclic codes and skew cyclic codes. The fact that any left $G_{(m,3,r)}$ -code is a direct sum of concatenated codes with inner codes ${\mathcal{ A}}_{i}$ and outer codes $C_{i}$ is proved, where ${\mathcal{ A}}_{i}$ is a minimal cyclic code over $\mathbb {F}_{q}$ of length $m$ and $C_{i}$ is a skew cyclic code of length 3 over an extension field of $\mathbb {F}_{q}$ . Then, an explicit expression for each outer code in any concatenated code is provided. Moreover, the dual code of each left $G_{(m,3,r)}$ -code is given and self-orthogonal left $G_{(m,3,r)}$ -codes are determined.