It depends on what you are doing, but I think the best plan might be to take trains at times, stay put for a while, and strategically rent cars for a week here or there. Thats what we have done in the past in Austria, France, and Italy.
I don't agree with Emily at all. You can not easily get to everywhere you want without a car. Try going to the famous resort of Hallstatt. You need to get off the train, then wait for a boat to take you across the lake. Driving there was much more convenient. And those boats did not seem to be moving very frequently. In fact, everything looked pretty dead after 6 pm, and god help you, if you arrived at 6:30 and had to spend a night in a train station.
If you check DB bahn, you'll see that taking trains via Hallstatt, Vienna, Krems An Donau, Salzburg, etc are not particularly convenient at all. It might take 2.5 hours from Vienna to Krems but only an hour driving. And taking a train from Linz to Cesky Krumluv was not piece of cake - its a 3.5 slow train that stops ever 10 minutes, and then you get dumped out at a closed train station and have to walk a mile into town.
If you buy a car, its going to cost $5k-10k for a used car, you have to worry about registration (possibly without having a permanent address in your host country), registration fees, possible smog certifications if its like the US. And the biggest worry is how to get rid of the car when you have a week left? Austrian Craigslist? And what if it doesn't sell? Are you going to lose your entire investment? Thats too big of a risk for me.