In this paper, we investigate serial concatenation of low-density parity check (LDPC) codes and minimum shift keying (MSK) with iterative decoding. We show that the design of LDPC codes is crucially dependent on the realization of the MSK modulator. For MSK modulators with non-recursive continuous phase encoders (CPEs), optimal codes for BPSK are optimal whereas for MSK modulators with recursive CPEs the BPSK codes are not optimal. We show that or nonrecursive CPEs, iterative demodulation and decoding is not required even though the CPE has memory. However, iterative demodulation is essential for recursive CPEs. For recursive CPEs, we design LDPC codes using density evolution and differential evolution considering iterative demodulation and decoding. The resulting codes provide significantly improved performance over the existing codes.