Motivated by applications in unsourced random access, this paper develops a novel scheme for the problem of compressed sensing of binary signals. In this problem, the goal is to design a sensing matrix $A$ and a recovery algorithm, such that the sparse binary vector $\mathbf{x}$ can be recovered reliably from the measurements $\mathbf{y}=A\mathbf{x}+σ\mathbf{z}$, where $\mathbf{z}$ is additive white Gaussian noise. We propose to design $A$ as a parity check matrix of a low-density parity-check code (LDPC), and to recover $\mathbf{x}$ from the measurements $\mathbf{y}$ using a Markov chain Monte Carlo algorithm, which runs relatively fast due to the sparse structure of $A$. The performance of our scheme is comparable to state-of-the-art schemes, which use dense sensing matrices, while enjoying the advantages of using a sparse sensing matrix.