Is it okay these days to use PNG with alpha-transparency in websites?

Yes it's absolutely fine to use PNG's with Alpha Transparency. It actually always has been, you just had to fix IE6 with filters.

So go ahead and use them :)

When I have to use alpha-transparent PNGs, I like to handle older browsers with DD_belantedPNG.

Then you can set up an IE conditional to call the PNG fix on the specific elements you need.

    <!--[if IE 6]>
        <script type="text/javascript" src="/lib/js/DD_belatedPNG-min.js"></script>
        <script type="text/javascript">
            DD_belatedPNG.fix('#Logo a img');
        </script>
    <![endif]--> 

You probably want to avoid blanket PNG fixes that apply to all PNGs on the page.

Remember, if you don't need the alpha-transparency 8-bit PNGs will work everywhere.