We develop a novel three-dimensional (3D) cluster-based channel model for vehicle-to-vehicle (V2V) communications under the scenarios of urban street scattering environments. The proposed model combines the flexibility of geometrical channel models with the existing state-of-the-art 3D V2V models. To provide an accurate representation of specific locations and realistic V2V fading environments in a computationally manageable fashion, all clusters are divided into three groups of use cases including "ahead," "between," and "behind" clusters according to the relative locations of clusters. Using the proposed V2V model, we first derive the closed-form expressions of the channel impulse response (CIR), including the line-of-sight (LoS) components and cluster components. Subsequently, for three categories of clusters, the corresponding statistical properties of the reference model are studied. We additionally derive the expressions of the 3D space-time correlation function (STCF), the autocorrelation function (ACF), and 2D STCF. Finally, comparisons with on-road measurement data and numerical experiments demonstrate the validity and effectiveness of the proposed 3D cluster-based V2V model.