Thanks for all the replies. I think it's something I'll just have to try and see how comfortable I feel. As people have suggested, I'll try out short tours at first, and go from there. I'll soon get a feel for how confident and comfortable I am.
I understand what slowmo's getting at, and I think he has a valid point. When I travelled in the Middle East, I dressed very conservatively, and carried a headscarf with me because I discovered very quickly that I could make myself "invisible" to men by covering my head. I was generally ignored by men unless I made eye contact when I was wearing the headscarf, but when I wasn't wearing it I did get a bit of unwanted attention - not in a bad way, usually just trying to sell me something, but it sometimes took a while to make it clear I wasn't interested. I'm not sure how this would translate to travelling in Europe, though. Well, let's face it, I'm not going to be particularly desirable to anyone who gets within a few metres of me if I've been cycling for a few days without any proper laundry facilities!
I have thought about getting a dog - I'd really like a dog anyway - but I know travelling with a dog brings its own complications. I may think more about that because it would be nice to have a companion, both at home and out on the bike, and the extra difficulties could be worth it.