If everything is working correctly, the difference should be subtle and you shouldn't generally notice a big shift.

I have a suspicion:

You may be working on a monitor which is not capable of rendering the whole Adobe RGB gamut. In this case, out-of-gamut colors are clipped or approximated (perhaps poorly). When you convert to sRGB, the colors are mapped more correctly and can be actually rendered, and you get the shift. In other words, the sRGB version was "right" all along — you just weren't seeing it.

This is one reason I recommend that most people work in sRGB even though bigger color spaces sound better.

The other possibility, if you're working in a nicely color-calibrated setup, is that your image does have a lot of tones not represented in sRGB and losing them does happen to make a big difference in the scene. In that case, there's nothing you can really do, although if web and sRGB are a large portion of your display audience you may want to take the time to do sRGB-specific color work.

This is kind of web-browser specific, but take a look at Microsoft's color management test page, and in particular the "image test" link. In a properly-configured system, as you click between Adobe RGB, sRGB, and ProPhoto (click between the or thises), you'll see only very subtle changes. If your environment isn't set up correctly (in the browser test, including if you don't have proper browser support), the Adobe RGB and ProPhoto examples will look horribly washed out and wrong. (Viewing the page on my iPhone provides a great example of a browser with no proper color profile support.)