This paper proposes a secure state estimation scheme with asynchronous non-periodic measurements for con-tinuous LTI systems under false data attacks on measurement transmission channels. Each sensor transmits the measurement information in a triple comprised of its sensor index, the time-stamp, and the measurement value to the fusion center via unprotected communication channels. A malicious attacker can corrupt a subset of sensors by (i) manipulating the time-stamp and the measurement value, (ii) blocking transmitted measurement triples, or (iii) injecting fake measurement triples. To deal with such attacks, we propose a secure state estimator by designing decentralized local estimators and fusing all the local states by the median operator. The local estimators receive the sampled measurements and update their local state in an asynchronous manner, while the fusion center triggers the fusion and generates a secure estimation in the presence of a local update. We prove that local estimators of benign sensors are unbiased with stable error covariance. Moreover, the fused secure estimation error has bounded expectation and covariance against at most p corrupted sensors as long as the system is 2p-sparse observable. The efficacy of the proposed scheme is demonstrated through a benchmark example of the IEEE 14-bus system.