The introduction of stochastic methods in pedigree analysis has enabled geneticists to tackle computations intractable by standard deterministic methods. Until now these stochastic techniques have worked by running a Markov chain on the set of genetic descent states of a pedigree. Each descent state specifies the paths of gene flow in the pedigree and the founder alleles dropped down each path. The current paper follows up on a suggestion by Elizabeth Thompson that genetic descent graphs offer a more appropriate space for executing a Markov chain. A descent graph specifies the paths of gene flow but not the particular founder alleles traveling down the paths. This paper explores algorithms for implementing Thompson's suggestion for codominant markers in the context of automatic haplotyping, estimating location scores, and computing gene-clustering statistics for robust linkage analysis. Realistic numerical examples demonstrate the feasibility of the algorithms.