It's as easy as calling AutoEurope or Kemwel or Hertz or Sixt or any of the other companies or brokers that do European rentals. You don't need anything other than your own license unless you're planning to rent a truck or motorcycle or something odd like that. There are virtually no borders anymore in Europe - you just drive from one country to the next.

That said, what you do NOT want to do is rent a car in one country and drop it off in another. You'll incur substantial fees for that.

You may have some difficulty driving a car into the Czech Republic; ask at the agencies cited in the above response

The Mosel Valley between Luxembourg, Trier and Koblenz Germany is IMO one of Europe's very most scenic drives - right along the sinuous river which runs in a narrow gorge with walls several hundred feet high - oft covered with vineyards that yield the celebrated local Mosel white wines.

Lots of cutesy small wine towns en route. Check out the Haus Lippman in Beilstein who everyone on Fodor's whose stayed there just raves about - also at one of the most gorgeous parts of the Mosel. Check out Burg Eltz also - high above the Mosel it is own of Germany's most celebrated castles.

Can I suggest you take the train into Germany from Amsterdam, and rent your car there. You can then drop it off in Munich, so you get to enjoy most of your planned drive, but avoid the huge one way inter country drop off fees. I am assuming you are not planning on returning to the Netherlands after Prague.
I would drive down the Mosel valley to Trier rather than going through Luxembourg. you could do a quick trip in and out of Luxembourg from Trier but the Mosel valley really is worth missing Luxembourg for.

Out of curiosity how much do the fees normally run?

Around $500, if you're referring to the fees for dropping off in a second country.

I agree with hetismij. We did sort of a reverse - took the train from AMS to Munich (about 8 hours if memory serves), and then did Munich and Neuschwanstein, Rothenburg ob der Tauber, Burg Eltz (staying in Beilstein), Trier, and then dropping the car in Koblenz, and taking the train back to AMS.
One of our favorite trips, and for whatever reason, we seem to use every available source of transportation on our trips, lol!

Also-- they may be funny about taking the car and what kind of car they give you-- no joke, in Munich, they asked us no leass than 5 times if we were going to drive into Italy. I know Italy has a higher theft rate, but it was almost comical.

Are you driving into Italy? "No"
Are you sure you are not driving to Italy? "Yes. We will only be in Germany"
It's ok if you want to go to Italy, I just have to give you a different car. "OK. But we are still not going to Italy."
Are you going to be driving anywhere outside of Germany?

One thing about driving a hire car from the Netherlands in Germany is it probably won't have an environment sticker to enable you to take the car into some German towns and cities. A German rental is far more likely to have one (though it may not of course).

The drop off fee depends on the company, and on many different factors. It may not be quoted if you try booking online and may be charged when you collect the car - it will say somewhere in the small print about it though.

I agree with the warnings about the expenses of cross-order drop-offs, and the limitations that some car rental firms impose on the make and model of car they will rent you for taking it to Italy and Prague.

Additionally, several ountries require that you have an international Driving Permit, in conjunction with your regular driver's license. Essentially, this is a translation of the data on your license into umpteen languages. In most countries that require them, you will never be asked for the Permit, EXCEPT if you have incurred a travel violation or have an accident. So, it's worth to spend the small amount they cost.

Where to get the permit? I don't know where you live, but here in the States we can get it for about US$10 at any Automobil Club office, right at the spot. Depending on your Club membership statyus, they may or may not charge you extra for the two passport pix you will need.

Another thing to consider:

You're crossing three borders:
Netherlands to Germany (as hetismij says, better to cross by train, and pick up in Germany, to avoid car dropoff fees);
Germany to Austria (do you just want to see Salzburg or do you want to go onto Vienna? Same thing applies: better to cross the border by train, then pick up car and head to Vienna. Drop off there, before crossing
Austria to Czech Rep. (Prague) by train.

You can easily take the train from Munich to Salzburg to Vienna, then to Prague, or return to Munich from Salzburg and thence on to Prague by train the whole way.

Go to bahn.de, it has a great website for checking train schedules all over Europe, not just Germany. (There's an English version; just click where indicated on an English-speaking country.)

We have done road trips all over europe. they are very easy - you just need to make sure of a couple of things:

1) you tell the rental agency in advance what countries you will be going into (drop off prices will differ and sometimes more luxurious cars are limited to certain countries)

2) make sure the car has the necessary insurance papers (1 set of idiots in Rome sent us off without the proper card and we had to circle back 10 minutes later). We now do a
pre-flight" check list - including that all of the lights, wipers work and we know how to get into and close the gas tank

3) Certain countries - including I believe Austria - require a tag on the car to pay for highways - be sure they give you one

4) Make sure you have really good maps - Michelin for preference for the larger areas and downloaded street maps for every town you will drive through

If you want to do a road trip - just do one. Yes, it might be cheaper to pick up and drop off cars in various places and coordinate with train schedules - but where is the romance of a road trip? The serendipity of seeing a cute village and just turning off to visit it. finding a pretty view and pulling over to take some pix? Finding an adorable restaurant yo can;t resist?

There isn't much hassle if you want to drive inter country if the car rental company allows you to do so. All you need to do is specify everything to dealer (travel agent). Also you might need to pay some extra money for traveling accros other border and you are all set to drive. http://www.pointcarhire.com/ is one such car rental comapny which I have personally used and is very customer friendly. Also they allow traveling across borders in Europe. And as per yours need you can hire car for longer duration so you may be eligible for some good discounts.

Certainly most car rental companies in the Netherlands will allow cross border rentals and will provide the green card with the vehicle. I am not against a road trip - I love road trips, but I am suggesting that the cross border drop off fee will buy a lot of petrol for a car and petrol is not cheap in Europe.

One thing I am a bit confused by is that you seem to want to visit Luxembourg and Trier then overnight in Heidelberg all in one day or am I missing something?
If that is your plan you need to rethink it.

Munich needs an Umweltsticker (environmentsticker) for you to take the car into the centre, as does Heidelberg I believe. So any car you hire must already have the sticker on it or you will risk a fine.