<p>Testing the homogeneity of means across independent count populations is a well-explored challenge in the literature. Traditional test procedures generally assume that the counts follow negative binomial, Poisson, or binomial distributions, which are suitable for over-, equi-, and under-dispersed counts, respectively. This article presents new test procedures for comparing the homogeneity of means among over-, equi-, and under-dispersed counts within a unified framework using the Conway-Maxwell-Poisson (CMP) distribution. The test statistics for the proposed procedures are developed using chi-square, conditional, unconditional, and modified analysis of variance (ANOVA) approaches. Two novel distributions, namely the generalized Conway-Maxwell Poisson and the generalized Conway-Maxwell multinomial distributions, are introduced for deriving the test statistic in the conditional testing approach. Additionally, a variance stabilizing transformation for the CMP distribution is applied in the modified ANOVA test. The effectiveness of the proposed test statistics is assessed through a simulation study, focusing on the empirical level and power. Finally, the application of these tests is illustrated with real data examples.</p>