We extend the Tobit (censored) linear equation system procedure to utility-theoretic demand functions, along with a mapping mechanism to impose the adding-up restriction implied by consumer utility maximization theory—a theoretical restriction very often ignored in previous empirical studies with censored demand systems. In this context, the Bayesian Markov Chain Monte Carlo (MCMC) procedure is applied to the censored linear approximate almost ideal demand system (LAIDS) for 12 food products, using data from the Turkish Household Expenditure Survey. All own-price elasticities are negative and expenditure elasticities positive. Uncompensated own-price elasticities for rural households are generally much higher than those for their urban peers, though demand for most food products are own-price elastic for both types of households. The differential patterns in demand elasticities between urban and rural households become even more evident (almost twice) as relatively more expensive foodstuffs are consumed, showing that accessibilities to alternative products have made rural Turkish households more cognizant toward price changes in foods. Household characteristics play a key role in food expenditures, notably so in urban areas, and regional and seasonal differences are also present.