MBA is definitely not the only way. MBB companies love people with non-standard careers. The companies are a bit selfish here, since they will show these people on every event as examples of the career transitions.
The best approach here will be the networking directly with the partners (ideally the partner responsible for recruiting). Here are some tips on how you can do that:
1) Leverage your own network - find friends or friends of friends who work at MBB and who can introduce you to the partner responsible for recruiting as a person with a non-standard background
2) Attend company events. Consulting companies do a lot of events both for graduates and experienced hires. Find the social network groups related to consulting or websites, subscribe to newsletters and stay tuned. While some of the events will be open to everyone, others will require a resume and a cover letter, so make sure to prep.
3) Talk to partners on the events and send Thank You notes. After each event, there is a Q&A session where you can talk to consultants 1 on 1 or in a group. Ask for contact info or send a thank you note after the event ("I just wanted to thank you for visiting our University... It was especially interesting to hear about... Would be happy to keep in touch and apply in the nearest future.). Alternatively, you may use linkedin for that. If you are an experienced hire I strongly suggest to ask for a 1 on 1 meeting in a thank you note.
4) Asking for advice. All people like giving an advice. So don't hesitate to ask consultants for an advice about your career change. Tell your story and ask how consulting fits into it