Graphs are widely used to model interconnected entities and improve downstream predictions in various real-world applications. However, real-world graphs nowadays are often associated with complex attributes on multiple types of nodes and even links that are hard to model uniformly, while the widely used graph neural networks (GNNs) often require sufficient training toward specific downstream predictions to achieve strong performance. In this work, we take a fundamentally different approach than GNNs, to simultaneously achieve deep joint modeling of complex attributes and flexible structures of real-world graphs and obtain unsupervised generic graph representations that are not limited to specific downstream predictions. Our framework, built on a natural integration of language models (LMs) and random walks (RWs), is straightforward, powerful and data-efficient. Specifically, we first perform attributed RWs on the graph and design an automated program to compose roughly meaningful textual sequences directly from the attributed RWs; then we fine-tune an LM using the RW-based textual sequences and extract embedding vectors from the LM, which encapsulates both attribute semantics and graph structures. In our experiments, we evaluate the learned node embeddings towards different downstream prediction tasks on multiple real-world attributed graph datasets and observe significant improvements over a comprehensive set of state-of-the-art unsupervised node embedding methods. We believe this work opens a door for more sophisticated technical designs and empirical evaluations toward the leverage of LMs for the modeling of real-world graphs.