If the UK would sign up to the Schengen agreement, you'd have no problems at all.
Belgium, France, Germany, Luxembourg, Monaco, the Netherlands, Portugal, Spain, Italy Austria, Greece,Denmark, Finland, Iceland, Norway and Sweden already have a 'no borders' policy. (i.e. anyone who's already in one country, irrespective of their own nationality, is free to move to any other country without any passport controls). During the coming year, the Czech Republic, Estonia, Hungary, Latvia, Lithuania, Malta, Poland, Slovakia and Slovenia will also 'join the club'.
Switzerland, Bulgaria, Cyprus and Romania are already committed to joining the Schengen territories at a later date.
The only EU member state which refuses to join the Schengen group is the UK. (The Republic of Ireland wishes to join but is prevented from doing so because of the existing 'open borders' policy with the UK). Regrettably, our Europhobic media prevent our politicians from being brave enough to sign up to Schengen (or, indeed, to replace the pound with the euro).
Similarly, the fluctuating views of the media prevent many MPs from backing national identity cards. While I'm one of those who think that the costs of such cards probably outweigh the benefits, it has to admitted that one such benefit is that any EU citizen who holds a nationality identity card can use it, instead of a passport, to visit any EU country.
So it's not EU law which prevents you from travelling without a passport. It's our politicians (and our media) who're out of step with the rest of Europe.
Chris