Science papers are surprisingly conformist. If you want to get published, you do it the way everybody else does. If you want to push Bayesian analyses, you are probably better off doing it alongside p values, instead of as a replacement for them.
Do bear in mind, though, that p-values and ANOVAs aren't wrong. They're specialized tools that researchers tend to misuse. Having a Bayesian background should help you understand what they are and aren't suitable for.
LWers may find useful two recent articles summarizing (for cognitive scientists) why Bayesian inference is superior to frequentist inference.
Kruschke - What to believe: Bayesian methods for data analysis
Wagenmakers et al - Bayesian versus frequentist inference
(The quote "Friends do not let friends compute p values" comes from the first article.)