Are there any legal issues with publishing part of an unpublished thesis in a journal? I have recently graduated and I was wondering whether, after submitting my thesis, there are any issues (legal or otherwise) if I publish parts of my thesis as a paper in a journal.

My thesis was not published as book, as was the case in this related question.
Ask your thesis advisor!  In most cases, the answer will be, "No.  Go for it!"

Schools love for their graduates to have publications.

In some institutions, including the one at which I teach, the institution claims ownership in the intellectual property of student work, but even then, a derivative work that results in a publication is likely to be welcome.