This paper proposes to construct a class of multiple-burst-correction quasi-cyclic (QC) codes via matrix transformations. Due to the diagonal structure of the transformed parity-check matrix of a QC code, multiple bursts can be corrected in the transform domain. By well designing the diagonal submatrices, the constructed QC codes are able to achieve optimal or asymptotically optimal multiple-burst-correction capability. In particular, a subclass of our constructed QC codes are QC low-density parity-check (QC-LDPC) codes which also perform very well over random channels. Simulation results show that our QC-LDPC codes outperform the existing QC-LDPC codes over burst channels.