This paper introduces a multiband spectrum sensing technique that utilizes nonuniform stratified sampling. Capitalising on the ability of the sampling scheme to suppress aliasing, the proposed method accomplishes the sensing task using sampling rates well below the ones demanded by the approaches based on uniform sampling. This effectively eases the stringent sampling rate requirements on the data acquisition module, especially when the spectrum sensing is conducted over wide bandwidths. The statistical characteristics of the adopted periodogram-type spectral analysis tool are examined and subsequently employed to formulate a dependable detection procedure amid a specified system performance. Recommendations are provided to ensure that the proposed technique satisfies the sought detection probabilities. These guidelines address the trade-off between the required sampling rate and the length of the signal observation window (sensing time) in a given scenario, offering the user the means to evaluate the benefits/advantages of the introduced approach. Numerical examples are presented to demonstrate the usefulness and effectiveness of the method. It is illustrated that stratified sampling can be implemented in practice using finite resources. This is a clear advantage over previously reported randomized sampling schemes such as total random sampling and Poisson sampling where the sampling instants can be arbitrarily close, i.e. demand infinitely fast acquisition device(s).