Let 𝓈 be a finite or countable set. Given a matrix F = (F
ij
)
i,j∈𝓈
of distribution functions on R and a quasistochastic matrix Q = (q
ij
)
i,j∈𝓈
, i.e. an irreducible nonnegative matrix with maximal eigenvalue 1 and associated unique (modulo scaling) positive left and right eigenvectors u and v, the matrix renewal measure ∑
n≥0
Q
n
⊗ F
*n
associated with Q ⊗ F := (q
ij
F
ij
)
i,j∈𝓈
(see below for precise definitions) and a related Markov renewal equation are studied. This was done earlier by de Saporta (2003) and Sgibnev (2006, 2010) by drawing on potential theory, matrix-analytic methods, and Wiener-Hopf techniques. In this paper we describe a probabilistic approach which is quite different and starts from the observation that Q ⊗ F becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate Q ⊗ F to a Markov random walk {(M
n
, S
n
)}
n≥0 with discrete recurrent driving chain {M
n
}
n≥0. It is then shown that renewal theorems including a Choquet-Deny-type lemma may be easily established by resorting to standard renewal theory for ordinary random walks. The paper concludes with two typical examples.