It depends. For a simple design with complete balance power could be similar to ANOVA, but for more complicated designs it won't be. There are some short-cut approaches but for anything complicated the best solution is usually to use simulations.

I believe the shortcuts Thom Baguley referred to involve estimating sample size as you would if the data did not have a multilevel structure, and then applying a correction factor. The attached image shows a page from Jos Twisk's introductory book on multilevel models that gives some more info about that. HTH.

Twisk, J. W. (2006). Applied multilevel analysis: a practical guide for medical researchers. Cambridge university press.

You can use simulation or analytic methods to calculate the statistical power of a mixed linear model. Here are the general steps for both approaches:

Both methods require knowledge of the parameters of the mixed linear model, including the variance components, which can be estimated using maximum likelihood or restricted maximum likelihood methods. Additionally, various software packages are available to calculate the statistical power of mixed linear models, such as SAS, R, and SPSS.