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This paper introduces a class of methods called TES (Transform-Expand-Sample) for generating autocorrelated variates with uniform marginals and Markovian structure. TES methods are readily implemented on a computer and have generation complexity comparable to that of the i.i.d, uniform sequence which they transform to an autocorrelated uniform sequence. For any prescribed correlation coefficient p, there is a IES method generating a uniform sequence with the 1-lag autocorrelation rho, and the resultant autocorrelation is monotonic quadratic in two structural TES parameters. A simulation study reveals that TES methods give rise to autocorrelation functions with monotone decreasing as well as oscillating magnitude, bounded by monotone envelopes. The structural parameters were found to control the "amplitude" and "frequency" of the resultant autocorrelation function. A third parameter can be used to transform a TES sequence into more continuous-looking versions and to control the skewness of sample path cycles. [ABSTRACT FROM AUTHOR]