Hey megandh, welcome to the forums!
Xilios has already covered some some of the major areas, so all I have to add is a couple of little points.
I also think hybrid bikes are a good, cheap alternative. I've seen people doing long tours on heavy iron mountain-beasts, and they managed just fine. Bring a small repair kit if you feel you'll be able to use it. Take the time to get a basic knowledge of simple repairs. It's a simple machine, even taking your bike apart to box it, and then putting it back together again will teach you a wealth of things about bike mechanics. A caveat with hybrid/used mountain bikes is that the smaller your wheel diameter, the heavier your bike and wider your tires, the more work it'll take you to do the distance on a tour. If you're up for the challenge and don't mind a little extra work though, that's no problem! You'll end up burning more calories and being able to eat more delicious European pastries that way

There's a difference however between working your body and hurting your body. The bike you get must be properly sized to accomodate you. Remember, you'll be in that saddle for hours every day, for months. Little strains and dicomforts you may not even notice for the few hours on the bike can accumulate and cause you pain, or even injury, in the long run. Take your bike to a good bike shop and ask them to size it for you (seat height, stem height etc). Tell them you plan on doing a long tour. Takes your bikes on a test ride where you bike for a weekend with the same load and the same distance that you plan to do at the start of your tour. Pay attention to what your body's telling you - are your knees/lower back/wrists screaming at the end of the ride? Research how you should prevent injury because of this, ask your physiotherapist, doctor or an experienced cyclist. If your handlebars can accomodate them, consider getting Aero bars for your bike. My biking partner and I didn't and I ended up learning to balance my elbows on my flat handlebars two months into our 4 month trip just to vary my body position. I highly, highly recommend them. The other alternatives are getting a bike with drop bars, or using bar ends like Xilios suggested.
Don't skimp on the tires, just to reiterate Xilios's suggestion. Cheap tires like ours blew (not due to puncture, just discintegrated) 1500 km into the trip under their load. The good ones are worth the expense. Getting puncture-proof tires or not is a personal choice. Ours were not, and I think I ended up fixing about 5 flats/2000km/bike, which wasn't too bad. Always carry a spare tube or two.
If you've got any more questions, feel free to post them on these forums. I can't emphasize enough how much the folks here helped me out when I was in your shoes.
Have a fantastic time planning, your 5 month tour sounds like the trip of a lifetime!
Cheers,
Blackbird