  Let $G_{(m,3,r)}=\langle x,y\mid x^m=1, y^3=1,yx=x^ry\rangle$ be a metacyclic
group of order $3m$, where ${\rm gcd}(m,r)=1$, $1<r<m$ 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 ${\rm gcd}(m,q)=1$ and
$r\equiv q^\epsilon$ 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 ${\cal A}_i$ and outer codes $C_i$ is proved, where ${\cal A}_i$ 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.
