We present a mesoscopic model for bitumen and bituminous mixtures. The model, which is based on dissipative particle dynamics, consists of different dynamical entities that represent the different characteristic time scales. Through the stress relaxation function, the mechanical properties of the model are investigated. For pure bitumen, the viscosity features super-Arrhenius behavior in the low-temperature regime in agreement with experimental data. The frequency-dependent viscoelastic properties show purely viscous behavior at low frequencies with increasing elasticity and hardening at higher frequencies, as expected. The model dynamics are analyzed in the framework of longitudinal hydrodynamics. The thermal process is two orders of magnitude slower than the attenuation of the density-wave propagation; hence the dynamic structure factor is dominated by a sharp Rayleigh peak and a relatively broad Brillouin peak. The model is applied to study triblock-copolymer-modified bitumen mixtures. Effects of the polymer concentration and end-block interactions with the bitumen are investigated. While the polymer concentration has an effect on the mechanical properties, the effect of increasing repulsive interactions between the bitumen and the polymer end-blocks is much more dramatic; it increases the viscosity of the mixture and shifts the onset of the elastic behavior to lower frequencies. For increased repulsion, the polymer end-blocks form small clusters that can be connected by a dynamic polymer backbone network. A simple Flory-Huggins analysis reveals the onset of segregation of the end-blocks in the bitumen mixture in agreement with the simulation data. Hence the changed mechanical properties are due to the emergence of large-scale structures as the repulsion is increased, which conforms to known mechanisms of microphase separation in polymer-modified bitumens.