It is possible to make a theme that does not use tables. bbPress is very flexible with themes. I am not aware if there exists one though…
But it uses tables by default because the list of topics and forums is in fact tabular data. I think it has nothing to do with table-based web-design of the 90’s (if that’s your concern).
I’ll upload a new version in a few weeks if you’re interested. Right now I have it installed in http://foros.ryuuko.cl (but adapted for the Thesis theme in WordPress).
Creating a theme without a table is easy enough, just remove the tables from the theme. However the question you need to ask is “why?”.
The default phpBB3 theme ProSilver uses a table-less layout and instead use definition lists to achieve a ‘table-like’ look, but most HTML/accessibility experts consider this to be an abuse of definition lists and do not recommend it. Tables are for tabular data and most people consider things like a board index to be in fact tabular, hence tables are typically used.