We have recently demonstrated 57 nm of tuning in a monolithic semiconductor laser using conventional DBR technology with grating elements removed in a periodic fashion. This paper describes the theory and design of these ''sampled'' grating tunable lasers. We first calculate sampled grating reflectivity. We then present normalized design curves which quantify tradeoff s involved in a sampled grating DBR laser with two mismatched sampled grating mirrors. These results are applied to a design example in the InP-InGaAsP system. The design example provides 70 nm tuning while maintaining > 30 dB MSR, with fractional index change DELTAmu/mu < 0.2% in the mirrors, and only 1 mm of total sampled grating length. Section IV summarizes recent experimental results, and compares them to theory. We also analyze other device structures which make use of sampled gratings.