Delay differential equations (DDEs) play an important role in the modeling of dynamic processes. Delays may arise in contemporary control schemes like networked distributed control and may cause deterioration of control performance, invalidating both stability and safety properties. This induces an interest in DDE especially in the area of modeling embedded control and formal methods for its verification. In this paper, we present an approach aiming at automatic safety verification of a simple class of DDEs against requirements expressed in a linear-time temporal logic. As requirements specification language, we exploit metric interval temporal logic (MITL) with a continuous-time semantics evaluating signals over metric spaces. We employ an interval-based Taylor over-approximation method to enclose the solution of the DDE. As the solution of the DDE gets represented as a timed state sequence in terms of the Taylor coefficients, we can effectively solve temporal-logic verification problems represented as time-bounded MITL formulae. We encode necessary conditions for their satisfaction as SMT formulae over polynomial arithmetic and use the iSAT3 SMT solver in its bounded model checking mode for discharging the resulting proof obligations, thus proving satisfaction of MITL specifications by the trajectories induced by a DDE. In contrast to our preliminary work, we can verify arbitrary time-bounded MITL formulae, including nesting of modalities, rather than just invariance properties.