Background
Although conventional parametric (and nonparametric) ANOVA require the assumption of equal variances, Welch (1951) developed an unequal variance oneway analysis of variance. This test has been little used to date, although its provision in R (as oneway.test) probably means it will be more heavily used in future.
Subsequent multiple comparison of means should be done using pairwise.t.test with a nonpooled standard deviation. This calculates pairwise comparisons between group levels with Bonferronitype corrections for multiple testing.
How to do it
Worked example
We will analyse the data from Johnston et al. (2001) using R:
Oneway unequal variance ANOVA

Gives something like this:
Oneway analysis of means (not assuming equal variances)
data: con and trt
F = 5.9763, num df = 2.000, denom df = 42.893, pvalue = 0.005134
Pairwise comparisons using t tests with nonpooled SD
data: con and trt
A B
B 0.0596 
C 0.0044 0.1280
P value adjustment method: holm