If you have coded specifically against IE 6 (working around issues, using non-standard features), it may be a little rough, but IE 8 is infinitely better in terms of stability and standards support.
Along with moving to IE 8, consider supporting other reasonably compliant browsers such as Firefox, Opera, Safari, or Chrome. IE 8 isn't a bad product, but other browsers will "keep you honest" and help you avoid hacks and also expose you to features not supported by IE 8.
It's also worth noting that pretty much any framework you may already be using (if it's still being maintained) will have equal or better support for IE 8. Same goes for any third party components which render HTML/script.
Lastly, I would also spend some time with IE 9 to understand the direction that Microsoft is going with the product. Most IE 8 apps will work fine with IE 9, but there are some subtle differences (mostly fixing/reinterpreting style and layout implementations in IE 8).
Edit: Here's an excellent, free tool I use for running different versions of IE side by side:
Here are several things off the top of my head that are non-standard IE extensions. These will probably still work in IE 8, but keep in mind that Microsoft is making a big push towards standards compliance, and they aren't guaranteed to work in the future. Also, other browsers don't support them at all.
Again, to my knowledge none of these things have been removed from IE 8 but they are all red flags for future maintainability.
@sahs - I would start with checking for the obvious things (if you haven't already) like invalid HTML and script errors. There's also a great free tool for running IE versions side-by-side that will let you test, fix, and compare results. I will update my post with the link.