Birth-and-death processes are widely used to model the development of biological populations. Although they are relatively simple models, their parameters can be challenging to estimate, because the likelihood can become numerically unstable when data arise from the most common sampling schemes, such as annual population censuses. Simple estimators may be based on an embedded Galton-Watson process, but this presupposes that the observation times are equi-spaced. We estimate the birth, death, and growth rates of a linear birth-and-death process whose population size is periodically observed via an embedded Galton-Watson process, and by maximizing a saddlepoint approximation to the likelihood. We show that a Gaussian approximation to the saddlepoint-based likelihood connects the two approaches, we establish consistency and asymptotic normality of quasi-likelihood estimators, compare our estimators on some numerical examples, and apply our results to census data for two endangered bird populations and the H1N1 influenza pandemic.