In user cooperative communications, time offsets among the relay nodes may distort the structure of a distributed space-time code, and therefore impair the diversity gain. In this paper, a distributed linear convolutive space-time coding scheme is proposed for asynchronous cooperative communication networks. We propose systematic construction methods of linear convolutive space-time codes (DLC-STC) that are .insensitive to the timing errors, i.e., they achieve the full spatial diversity under any delay profile. We then study the diversity property of the delay-tolerant DLC-STC with suboptimal receivers that have low decoding complexity. We show that with the ZF, MMSE and MMSE-DFE receivers, our proposed codes also achieve the full spatial diversity in the asynchronous scenario.