This contribution focuses on the use of the pole placement approach in the design of observers for certain classes of Linear-Time-Invariant (LTI) systems with time delay. Specifically, depending on how the delay appears in the system dynamics, two classes of dynamic systems are considered: state-delayed systems and input-delayed systems, respectively. First, we address the problem of designing a full-order Luenberger observer for the considered systems using partial placement of the error poles. Namely, we exploit the multiplicity-induced-dominancy (MID) property of the characteristic root with the maximal admissible nultiplicity of the characteristic function corresponding to the system's error. After giving the existing condition for the proposed observer, we use the same MID property but in the so-called generic case. The performance and effectiveness of the proposed observers are highlighted through several illustrative examples.