I have an "always on" Internet connection (cable, DSL, etc.), but my IP address is assigned dynamically. Can I still run my own DNS server?

You can run Simple DNS Plus and take advantage of DNS caching no matter what type of Internet connection you are using (even dial-up).

In order to host domain names with your own DNS server, you need an IP address which doesn't change (or changes very seldom).
Many cable/DSL Internet providers do assign IP addresses dynamically, but in most cases this does not mean that your IP changes - you just get the same IP address assigned over and over again.
So for all intents and purposes the IP address is static - and is perfect for running a DNS server hosting domain names.
To make sure that your IP address doesn't change, you probably need to monitor this for at least 10 days (or contact your connection provider).