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In this paper, we propose an Alamouti-code-based relaying scheme for frequency asynchronous amplify-and-forward (AF) relay networks. Both the oscillator frequency offsets and the Doppler shifts among the distributed nodes are considered in our design. We employ orthogonal frequency-division multiplexing (OFDM) modulation at the source node and let the two relay nodes implement only simple operations, such as time reversal, conjugation, and amplification. We show that without Doppler shifts, the multiple carrier frequency offsets (CFOs) can be directly compensated at the destination, and the received signals exhibit an Alamouti-like structure. We further prove that full spatial diversity can be achieved by the fast symbol-wise detection when the oscillator frequency offset between the relay nodes is smaller than a certain threshold, which yields lower decoding complexity compared with the existing schemes. In the case with Doppler shifts, where the direct CFO compensation becomes impossible, we develop a repetition-aided Alamouti coding approach, by which full diversity can be nearly achieved from the fast symbol-wise detection. Numerical results are provided to corroborate the proposed studies.