My Kindle (version 3) suddenly won't connect to my wireless network. I've re-entered our wireless passcode, but still doesn't work.

On a wifi connection when you see problems the cause will either be the device or the wifi router.
If you can try the kindle out on a different network this will tell you if it is the kindle or the network. If it connects then it not the kindle. It could be a setting or the router.

If it is a setting the two biggies are going to be something wrong with the security passcode or it isn't getting through the gateway out onto the internet. For settings you need to check things like using wep when it should be wpa or the wep key is 64bit when it needs to be 128 or in the wrong key location and so on.

When it can't get through the gateway that usually means your dhcp is wrong or you are using fixed ip. You can just set these numbers manually in the kindle.

The other usual suspect is that the router itself is beginning to fail. This usually shows up first as difficulty to connect and dropped connections or increasing errors. It will eventually fail. This is common and typical of all home grade wifi routers. They are good for about 18 months or so. If this is the case you might notice slow downs or dropped connections on other wifi devices. These units are not repairable so you just need to replace it.

Out of all the possibilities I would say your first choice would be the security settings and next would be the the router. Those are the two most common problems.

I found that with my doing a factory reset kindle 3 every so offen clears most of the buggy issues that arise after a few months of heavy use. I would say that a full factory reset would solve all software issues on any version Kindle. The downside is if you have uploaded your own ebooks (not purchased from Amazon) you most definetly will have to reload them. Not to mention settings etc. When all else fails try a factory reset works like a charm.