Authenticated key agreement protocol is a useful cryptographic primitive, which can be used to protect the confidentiality, integrity and authenticity for transmitted data over insecure networks. From the point of view of the management of pre-shared secrets, one of the advantages of three-party authenticated key agreement (3PAKA) protocols is that they are more suitable for use in a network with large numbers of users compared with two-party authenticated key agreement protocols. Using smart cards is a practical, secure measure to protect the secret private keys of a user. Recently, some 3PAKA protocols using smart cards have been proposed. However, up to now, it is still a challenging problem to propose a 3PAKA protocol using smart cards with fewer rounds of messages and without using timestamp technique. Another important fact to be mentioned is that existing 3PAKA protocols using smart cards all lack provable-security guarantees. In this paper, we propose a new 3PAKA protocol using smart cards. The proposed protocol gains several advantages over existing related protocols: (1) The protocol is provably secure under the computational Diffie-Hellman assumption in the random oracle model, and hence can resist strong adversaries in network scenarios; (2) The protocol needs fewer rounds of messages, and can enable short communication latency and rapid response; and (3) The protocol is not based on timestamp technique, and does not need the complicated clock synchronization.