We develop a novel class of tests to detect mitochondrial DNA (mtDNA)-mutation involvement in complex diseases by the study of affected pedigree members. For a pedigree, affected individuals are first considered and are then connected through their relatives. We construct a reduced pedigree from an original pedigree. Each configuration of a reduced pedigree is given a score, with high scores given to configurations that are consistent with mtDNA-mutation involvement and low scores given to configurations that are not consistent with mtDNA-mutation involvement. For many pedigrees, the weighted sum of scores of the pedigrees is calculated. The tests are formed by comparing the observed score with the expected score under the null hypothesis that only nuclear autosomal mutations are involved. We study the optimality of score functions and weights under the heterogeneity model without phenocopies. We also develop a method to estimate the contribution that mtDNA mutations make if they are involved under a heterogeneity model. Finally, we apply our methods to three data sets: Leber hereditary optic neuropathy, a disease that has been proved to be caused by mtDNA mutations; non-insulin-dependent diabetes mellitus (NIDDM); and hypertension (HTN). We find evidence of mtDNA-mutation involvement in all three diseases. The estimated fraction of patients with NIDDM due to mtDNA-mutation involvement is 22% (95% confidence interval [CI] 6%-38%). The fraction of patients with HTN potentially due to mtDNA-mutation involvement is estimated at 55% (95% CI 45%-65%).