The answer is the latter: Might as well do regular ANOVA. If the DV and the covariate are not correlated then there isn't much point to doing an analysis of covariance: the covariate will not usefully extract variance from the DV. After all, the analysis is based on the assumption that the covariate is introducing variability into to DV in such a way that those who score low on the covariate score low on the DV; moderate on the covariate = moderate on the DV; high on the covariate = high on the DV (or the inverse if the DV and covariate are significantly negatively correlated). That's what makes something a "covariate". So one reason the various treatment groups may differ on the DV is the effect of the covariate. The analysis of covariance partials this effect of the covariate on the DV out of the effect(s) that the IVs have on the DV, giving you a "purer" look at the IVs' unique effect(s).
In addition to checking whether the DV and covariate are correlated at all it is important to look at the DV/covariate bivariate correlation to make sure they are LINEARLY correlated. If they are curvilinearly related then the covariate will not remove variance from the DV in a consistent way across the treatment groups. If this is the case, choose a different covariate, one that is strongly and linearly correlated with the DV.