In the present paper we introduce a new class of contact manifolds. Such type manifolds are called almost contact $(k,m)$-spaces with cyclic-parallel $t$-curvature tensor. We investigate some curvature properties of these type manifolds and we obtain that these type of manifolds have cyclic parallel Ricci tensor under some conditions. Furthermore, we get these type of manifolds are $K$-contact under some special cases and we obtain that $M$ is locally isometric to the product $E^{(n+1)}\times S^n (4)$ under some algebraic conditions.